Department of Mathematics at UTSA - User contributions [en]

MAT1043

2022-04-15T16:36:18Z

Khanh: /* Introduction to Mathematics */

==Introduction to Mathematics==
[https://catalog.utsa.edu/search/?P=MAT%201043 MAT 1043. Introduction to Mathematics]. (3-0) 3 Credit Hours. (TCCN = MATH 1332)

Prerequisite: Satisfactory performance on a placement examination. This course is designed primarily for the liberal arts major to satisfy the Core Curriculum mathematics requirement. Topics may include logic; proofs; deductive and inductive reasoning; number theory; fundamentals of statistics; basic statistical graphs; causal connections; financial management; functions; linear graphs and modeling; exponential growth and decay; logarithms; fundamentals of probability; fundamentals of geometry; and basic ideas from trigonometry, calculus, and discrete mathematics. (Formerly MTC 1043. Credit cannot be earned for both MAT 1043 and MTC 1043.) May apply toward the Core Curriculum requirement in Mathematics. Generally offered: Fall, Spring, Summer. Course Fees: LRC1 $12; LRS1 $45; STSI $21.

{| class="wikitable sortable"
|-
! Date !! Sections !! Topics !! Prerequisite Skills !! Student learning outcomes
|-
| Week 1 || Lesson 1.A || [[Data]] || Understand what is a learning community
||
* Collect data from daily life
*Work positively in a group to make a decision
|-
| Week 1 || Lesson 1.B || [[Learning Community]] || || Seek and give help to one another inside and outside of class
|-
| Week 1 || Lesson 1.C || [[First-degree equation involving percentages]] || Determine the original amount, given the percentage that a given number is of the original
||
* Understand the difference between the terms plurality and majority in an election
* Create a first-degree equation involving percentages and solve for the variable
* Apply and justify selection strategies to election results and decisions about other issues
* Multiple ranking methods can be employed to make decisions about other issues
* Employ the "Instant Runoff" method to determine the winner of an election
|-
| Week 2 || Lesson 1.D || [[Ranking methods]] || ||
* Earning the most votes may not be sufficient to win an election
* Employ the Borda Count method to determine the winner of an election
* Apply and justify selection strategies to election results
|-
| Week 2 || Lesson 2.A || [[Graphical Display]]
||
* Be able to read and interpret graphical displays
* Know the definition of mean
* Know the definition of median
||
* Analyze a variety of graphical displays and interpret them in context
* Compute the mean of a set of data
* Construct a dot plot or histogram from data
|-
| Week 2 || Lesson 2.C || [[Analyzing graphical displays]]
||
* Compare side-by-side graphical displays
* Be able to read and interpret graphical displays
|| Write a contextual analysis of a graphic display in a formal paper including appropriate mathematical language and explanations
|-
| Week 3 || Lesson 3.A || [[Sampling]] || Students should know the symbols for pop. Mean, sample mean, pop. Standard deviation, sample standard deviation
||
* Explain the difference between populations and samples
* Use the characteristics of a sample to describe the population
* Analyze the conclusions of a study and explain the limitations on inferences made
|-
| Week 3 || Lesson 3.B || [[Mean and Central Limit Theorem]] || Students should know how to determine the mean of a data set || Graph sample means and use the central limit theorem to estimate the population mean
|-
| Week 3 || Lesson 3.C || [[Standard Deviation]] || Students should be able to calculate the size of a portion given the size of the whole and a percentage
||
* Use standard deviation to interpret the spread of a data set
* Calculate the percentage of data in a graph region
|-
| Week 4 || Lesson 4.A || [[Probability]]
|| Students should be able to convert between fractions, decimals, and percentages ||
* Calculate theoretical probability of two or more independent events
* Calculate AND and OR probabilities for independent events
|-
| Week 4 || Lesson 4.B || [[Conditional Probability]] || Students should be able to determine a conditional probability || Calculate conditional probabilities for two or more dependent events
|-
| Week 5 || Lesson 5.A || [[Conversions]]
||
* Be able to calculate unit ratios
* Be able to use ratios to convert units
* Be able to perform dimensional analysis
||
* Recognize when converting units is needed
* Use conversion to make comparisons
|-
| Week 5 || Lesson 5.B || [[Index numbers]]
||
* Be able to convert ratios to decimals and percentages, and divide
* Be able to describe types of averages
||
* Perform calculations involving index numbers
* Make and justify decisions and evaluate claims using index numbers
|-
| Week 5 || Lesson 5.C || [[Weighted averages]] || Should be able to calculate mean
||
* Calculate weighted averages
* Use weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 5.D || [[Expected Value]] || Be able to calculate, percentages, means, and weighted averages
be able to find the mean of a data set
||
* Calculate expected value
* Make predictions about real world scenarios
|-
| Week 6 || Lesson 6.A || [[Weighted moving average graphs]]
||
* Weighted moving average
* Graph points
* Find the mean of a data set
||
* Calculate and compare simple and weighted moving averages
* Analyze graphs of moving average data
* Use simple and moving weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 6.B || [[Weighted moving average graphs continued]]
||
* Weighted moving average
* Find the mean of a data set
* Ue a spreadsheet to do calculations and create graphs
||
* Calculate and compare simple and weighted moving averages
* Write a contextual analysis of a graphical display
* Write a ratio or percentage and explain its meaning within a context
|-
| Week 6 || Lesson 7.A || [[Ratios and percentages]] || Be able to write and simplify fractions, create a pie graph, convert fractions to percentages
||
* Determine percentages based on part-to-whole ratios
* Write a ratio or percentage and explain in meaning within a context
* Read a budget, determine values of line items, and draw conclusions about the overall distributions of funds
|-
| Week 7 || Lesson 7.B || [[Part-to-part ratios & Part-to-whole ratios]] ||
* Convert ratios to percentages
* Sketch pie graphs using part-to-whole percentages
|| Construct a pie graph based on ratios and percentages
|-
| Week 7 || Lesson 7.C || [[Absolute change (additive reasoning) & Relative change (multiplicative reasoning)]] || Students should be able to create a line graph from data
||
* Analyze data in a spreadsheet and graphs
* Develop reasonable hypothesis supported by evidence
* Create line graphs and describe patterns in graphs
* Analyze and describe data using absolute (additive) comparison and relative (multiplicative) reasoning
|-
| Week 7 || Lesson 7.D || [[Adjusting claims and hypothesis]] || Students should be able to create a line graph from data || Analyze data in a spreadsheet and graphs to compare changes in categories
|-
| Week 8 || Lesson 7.E || [[Debt-to-income (DTI) ratios]] || Be able to write ratios and proportions, solve proportions, calculate percentages from ratios
||
* Calculate a DTI ratio
* Draw conclusions from DTI about the appropriateness of the percentage of income spent on housing and debt
|-
| Week 8 || Lesson 7.F || [[Proportional reasoning]] || Write rates, convert ratios to percentages
||
* Interpret ratios and percentages as rates of change
* Compare ratios and percentages
* Interpret graphical displays
* Compare mathematical relationships using a variety of representations
|-
| Week 8 || Lesson 8.A || [[Mathematical (Linear) relationships]]
||
* Be able to plot ordered pairs, sketch graphs of linear equations, construct a linear equation given the slope and y intercept
* Write a linear equation based on a verbal description/or that passes through 2 points
* Solve a 2-step linear equation
||
* Compare mathematical relationships using a variety of representations
* Write a linear equation given a slope and y intercept
|-
| Week 9 || Lesson 8.B || [[Proportionality vs. Linearity]] || Determine when 2 quantities are proportional || Explain, compare, and contrast linear and proportional relationships
|-
| Week 9 || Lesson 8.C || [[Simple and Compound Interest (Linear and Exponential Models)]]
||
* Write a percentage as a decimal
* Perform calculations with percentages
||
* Describe the difference between simple and compound interest in practical and mathematical terms
* Compare and contrast patterns in linear and exponential models
|-
| Week 10 || Lesson 8.D || [[Regression]]
||
* Read a scatterplot
* Interpret the slope of a line in context
||
* Create a scatterplot and regression equation using technology and estimate the parameters of the line of best fit
* Interpret the parameters (slope, y-intercept, coefficient of determination) of a simple linear regression
|-
| Week 10 || Lesson 8.E || [[Piecewise Linear Function]]
||
* Use a percentage rate to calculate tax
* Use constant rates of change to write a linear equation
* Identify the slope of a linear function
* Solve a linear equation for a given output
||
* Model a progressive income tax system algebraically and graphically
* Compare a progressive income tax system to a flat tax system and explain advantages and disadvantages of different income tax systems
|-
| Week 10 || Lesson 9.A || [[Depreciation]]
||
* Graph ordered pairs
* Identify the value of the output variable, given the input value, using a graphical and symbolic representation of the relationship between two variables
||
* Interpolate and extrapolate using a graphical representation of the relationship between two variables
* Use a symbolic model to find the exact value of one variable, given the value of the other variable, and relate those values to the context of the problem
|-
| Week 10 || Lesson 9.B || [[Geometric interpretation of interpolation ]]
||
* Graph data from a scenario
* Create proportions using the sides of similar triangles and solve them
||
* Create a proportion between corresponding sides of similar triangles
* Use variables with subscripts
|-
| Week 11 || Lesson 9.C || [[Univariate, Bivariate, Correlation and Causation]]
||
* Calculate measures of central tendency
* Analyze data and visual displays of univariate and bivariate data and describe trends
||
* Create a line graph for univariate data
* Determine, informally, the correlation between bivariate data
* Analyze data and related graphs and describe the trend of the data
|-
| Week 11 || Lesson 9.D || [[Univariate, Bivariate, Correlation and Causation]] || Create and interpret a scatterplot || Explain why, even if there is a strong correlation, a change in one variable may not cause a change in the other
|-
| Week 11 || Lesson 10.A || [[Time series model of exponential growth]] || Use formulas in spreadsheets
||
* Develop a time series model for the Fibonacci problem
* Test whether data are exponential by comparing the rate of growth to the population size
|-
| Week 12 || Lesson 10.B || [[Linear and Exponential Models]]
||
* Create a table of values and scatterplot in a spreadsheet
* Calculate the average rate of growth for a period of time
* Calculate the first differences in a data set
||
* Evaluate the mathematical appropriateness of a model give historical data
* Determine whether a data set suggests a linear or exponential relationship
* Identify and use an appropriate model to predict a future outcome
|-
| Week 12 || Lesson 11.A || [[Continuous Growth]]
||
* Analyze the relationship of input and output values in a problem situation
* Determine whether numerical and graphical relationships are increasing at a constant rate, at an increasing rate or at a decreasing rate
||
* Sketch a model for a population that increases at an increasing rate and a decreasing rate
* Identify behavior in a graph, draw conclusions about the behavior, and predict future outcomes
|-
| Week 12 || Lesson 11.B || [[Carrying Capacity and Logistic Growth Rate]]
||
* Calculate absolute and relative change
* Read delta notation for absolute change
||
* Develop discrete models of natural phenomena and use the models to predict future values
* Calculate the carrying capacity and logistic growth rate of a real-world scenario
|-
| Week 13 || Lesson 11.C || [[Logistic Growth Model]] || Determine the increasing and/or decreasing behavior of outputs in a table || Explore the changes of the values of the parameters of a logistic growth model and describe the effect of those changes on the model
|-
| Week 13 || Lesson 11.D || [[Complex Population Growth and Decay Models]]
||
* Find the constant of proportionality and express variables that are jointly proportional
* Determine how jointly proportional variables affect each other in an abstract model
|| Develop a parameterized time series model with more than two dependent variables in a spreadsheet
|-
| Week 14 || Lesson 11.E || [[Analyzing Complex Population Growth and Decay Models]] || Extract data from an academic article and create models for the data
||
* Determine parameters to match a model’s predictions against historical data
* Create a spreadsheet involving the formulas of the model to predict future behavior
|-
| Week 14 || Lesson 12.A || [[Periodic Function]]
||
* Plot points on coordinate axis
* Analyze the shape of a graph and find 12A coordinate values on a graph
||
* Sketch a graph that depicts a periodic phenomenon
* Identify the period and amplitude of a periodic function
* Compare and contrast the graphs of different periodic models
|-
| Week 15 || Lesson 12.B || [[The Sine Function]]
||
* Plot points on a graph
* Understand that constants (Parameters) in an equation control the relationship between the dependent variable and independent variable
||
* Describe the effect that changing one or more parameters has on the graph of a sine function
* Change the parameters of the sine curve to match given criteria
|}

MAT1043

2022-04-15T16:33:11Z

Khanh: /* Introduction to Mathematics */

==Introduction to Mathematics==
[https://catalog.utsa.edu/search/?P=MAT%201043 MAT 1043. Introduction to Mathematics]. (3-0) 3 Credit Hours. (TCCN = MATH 1332)

Prerequisite: Satisfactory performance on a placement examination. This course is designed primarily for the liberal arts major to satisfy the Core Curriculum mathematics requirement. Topics may include logic; proofs; deductive and inductive reasoning; number theory; fundamentals of statistics; basic statistical graphs; causal connections; financial management; functions; linear graphs and modeling; exponential growth and decay; logarithms; fundamentals of probability; fundamentals of geometry; and basic ideas from trigonometry, calculus, and discrete mathematics. (Formerly MTC 1043. Credit cannot be earned for both MAT 1043 and MTC 1043.) May apply toward the Core Curriculum requirement in Mathematics. Generally offered: Fall, Spring, Summer. Course Fees: LRC1 $12; LRS1 $45; STSI $21.

{| class="wikitable sortable"
|-
! Date !! Sections !! Topics !! Prerequisite Skills !! Student learning outcomes
|-
| Week 1 || Lesson 1.A || [[Data]] || Understand what is a learning community
||
* Collect data from daily life
*Work positively in a group to make a decision
|-
| Week 1 || Lesson 1.B || [[Learning Community]] || || Seek and give help to one another inside and outside of class
|-
| Week 1 || Lesson 1.C || [[First-degree equation involving percentages]] || Determine the original amount, given the percentage that a given number is of the original
||
* Understand the difference between the terms plurality and majority in an election
* Create a first-degree equation involving percentages and solve for the variable
* Apply and justify selection strategies to election results and decisions about other issues
* Multiple ranking methods can be employed to make decisions about other issues
* Employ the "Instant Runoff" method to determine the winner of an election
|-
| Week 2 || Lesson 1.D || [[Ranking methods]] || ||
* Earning the most votes may not be sufficient to win an election
* Employ the Borda Count method to determine the winner of an election
* Apply and justify selection strategies to election results
|-
| Week 2 || Lesson 2.A || [[Graphical Display]]
||
* Be able to read and interpret graphical displays
* Know the definition of mean
* Know the definition of median
||
* Analyze a variety of graphical displays and interpret them in context
* Compute the mean of a set of data
* Construct a dot plot or histogram from data
|-
| Week 2 || Lesson 2.C || [[Analyzing graphical displays]]
||
* Compare side-by-side graphical displays
* Be able to read and interpret graphical displays
|| Write a contextual analysis of a graphic display in a formal paper including appropriate mathematical language and explanations
|-
| Week 3 || Lesson 3.A || [[Sampling]] || Students should know the symbols for pop. Mean, sample mean, pop. Standard deviation, sample standard deviation
||
* Explain the difference between populations and samples
* Use the characteristics of a sample to describe the population
* Analyze the conclusions of a study and explain the limitations on inferences made
|-
| Week 3 || Lesson 3.B || [[Mean and Central Limit Theorem]] || Students should know how to determine the mean of a data set || Graph sample means and use the central limit theorem to estimate the population mean
|-
| Week 3 || Lesson 3.C || [[Standard Deviation]] || Students should be able to calculate the size of a portion given the size of the whole and a percentage
||
* Use standard deviation to interpret the spread of a data set
* Calculate the percentage of data in a graph region
|-
| Week 4 || Lesson 4.A || [[Probability]]
|| Students should be able to convert between fractions, decimals, and percentages ||
* Calculate theoretical probability of two or more independent events
* Calculate AND and OR probabilities for independent events
|-
| Week 4 || Lesson 4.B || [[Conditional Probability]] || Students should be able to determine a conditional probability || Calculate conditional probabilities for two or more dependent events
|-
| Week 5 || Lesson 5.A || [[Conversions]]
||
* Be able to calculate unit ratios
* Be able to use ratios to convert units
* Be able to perform dimensional analysis
||
* Recognize when converting units is needed
* Use conversion to make comparisons
|-
| Week 5 || Lesson 5.B || [[Index numbers]]
||
* Be able to convert ratios to decimals and percentages, and divide
* Be able to describe types of averages
||
* Perform calculations involving index numbers
* Make and justify decisions and evaluate claims using index numbers
|-
| Week 5 || Lesson 5.C || [[Weighted averages]] || Should be able to calculate mean
||
* Calculate weighted averages
* Use weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 5.D || [[Expected Value]] || Be able to calculate, percentages, means, and weighted averages
be able to find the mean of a data set
||
* Calculate expected value
* Make predictions about real world scenarios
|-
| Week 6 || Lesson 6.A || [[Weighted moving average graphs]]
||
* Weighted moving average
* Graph points
* Find the mean of a data set
||
* Calculate and compare simple and weighted moving averages
* Analyze graphs of moving average data
* Use simple and moving weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 6.B || [[Weighted moving average graphs continued]]
||
* Weighted moving average
* Find the mean of a data set
* Ue a spreadsheet to do calculations and create graphs
||
* Calculate and compare simple and weighted moving averages
* Write a contextual analysis of a graphical display
* Write a ratio or percentage and explain its meaning within a context
|-
| Week 6 || Lesson 7.A || [[Ratios and percentages]] || Be able to write and simplify fractions, create a pie graph, convert fractions to percentages
||
* Determine percentages based on part-to-whole ratios
* Write a ratio or percentage and explain in meaning within a context
* Read a budget, determine values of line items, and draw conclusions about the overall distributions of funds
|-
| Week 7 || Lesson 7.B || [[Part-to-part ratios & Part-to-whole ratios]] || || Construct a pie graph based on ratios and percentages
|-
| Week 7 || Lesson 7.C || [[Absolute change (additive reasoning) & Relative change (multiplicative reasoning)]] || Students should be able to create a line graph from data
||
* Analyze data in a spreadsheet and graphs
* Develop reasonable hypothesis supported by evidence
* Create line graphs and describe patterns in graphs
* Analyze and describe data using absolute (additive) comparison and relative (multiplicative) reasoning
|-
| Week 7 || Lesson 7.D || [[Adjusting claims and hypothesis]] || Students should be able to create a line graph from data || Analyze data in a spreadsheet and graphs to compare changes in categories
|-
| Week 8 || Lesson 7.E || [[Debt-to-income (DTI) ratios]] || Be able to write ratios and proportions, solve proportions, calculate percentages from ratios
||
* Calculate a DTI ratio
* Draw conclusions from DTI about the appropriateness of the percentage of income spent on housing and debt
|-
| Week 8 || Lesson 7.F || [[Proportional reasoning]] || Write rates, convert ratios to percentages
||
* Interpret ratios and percentages as rates of change
* Compare ratios and percentages
* Interpret graphical displays
* Compare mathematical relationships using a variety of representations
|-
| Week 8 || Lesson 8.A || [[Mathematical (Linear) relationships]]
||
* Be able to plot ordered pairs, sketch graphs of linear equations, construct a linear equation given the slope and y intercept
* Write a linear equation based on a verbal description/or that passes through 2 points
* Solve a 2-step linear equation
||
* Compare mathematical relationships using a variety of representations
* Write a linear equation given a slope and y intercept
|-
| Week 9 || Lesson 8.B || [[Proportionality vs. Linearity]] || Determine when 2 quantities are proportional || Explain, compare, and contrast linear and proportional relationships
|-
| Week 9 || Lesson 8.C || [[Simple and Compound Interest (Linear and Exponential Models)]]
||
* Write a percentage as a decimal
* Perform calculations with percentages
||
* Describe the difference between simple and compound interest in practical and mathematical terms
* Compare and contrast patterns in linear and exponential models
|-
| Week 10 || Lesson 8.D || [[Regression]]
||
* Read a scatterplot
* Interpret the slope of a line in context
||
* Create a scatterplot and regression equation using technology and estimate the parameters of the line of best fit
* Interpret the parameters (slope, y-intercept, coefficient of determination) of a simple linear regression
|-
| Week 10 || Lesson 8.E || [[Piecewise Linear Function]]
||
* Use a percentage rate to calculate tax
* Use constant rates of change to write a linear equation
* Identify the slope of a linear function
* Solve a linear equation for a given output
||
* Model a progressive income tax system algebraically and graphically
* Compare a progressive income tax system to a flat tax system and explain advantages and disadvantages of different income tax systems
|-
| Week 10 || Lesson 9.A || [[Depreciation]]
||
* Graph ordered pairs
* Identify the value of the output variable, given the input value, using a graphical and symbolic representation of the relationship between two variables
||
* Interpolate and extrapolate using a graphical representation of the relationship between two variables
* Use a symbolic model to find the exact value of one variable, given the value of the other variable, and relate those values to the context of the problem
|-
| Week 10 || Lesson 9.B || [[Geometric interpretation of interpolation ]]
||
* Graph data from a scenario
* Create proportions using the sides of similar triangles and solve them
||
* Create a proportion between corresponding sides of similar triangles
* Use variables with subscripts
|-
| Week 11 || Lesson 9.C || [[Univariate, Bivariate, Correlation and Causation]]
||
* Calculate measures of central tendency
* Analyze data and visual displays of univariate and bivariate data and describe trends
||
* Create a line graph for univariate data
* Determine, informally, the correlation between bivariate data
* Analyze data and related graphs and describe the trend of the data
|-
| Week 11 || Lesson 9.D || [[Univariate, Bivariate, Correlation and Causation]] || Create and interpret a scatterplot || Explain why, even if there is a strong correlation, a change in one variable may not cause a change in the other
|-
| Week 11 || Lesson 10.A || [[Time series model of exponential growth]] || Use formulas in spreadsheets
||
* Develop a time series model for the Fibonacci problem
* Test whether data are exponential by comparing the rate of growth to the population size
|-
| Week 12 || Lesson 10.B || [[Linear and Exponential Models]]
||
* Create a table of values and scatterplot in a spreadsheet
* Calculate the average rate of growth for a period of time
* Calculate the first differences in a data set
||
* Evaluate the mathematical appropriateness of a model give historical data
* Determine whether a data set suggests a linear or exponential relationship
* Identify and use an appropriate model to predict a future outcome
|-
| Week 12 || Lesson 11.A || [[Continuous Growth]]
||
* Analyze the relationship of input and output values in a problem situation
* Determine whether numerical and graphical relationships are increasing at a constant rate, at an increasing rate or at a decreasing rate
||
* Sketch a model for a population that increases at an increasing rate and a decreasing rate
* Identify behavior in a graph, draw conclusions about the behavior, and predict future outcomes
|-
| Week 12 || Lesson 11.B || [[Carrying Capacity and Logistic Growth Rate]]
||
* Calculate absolute and relative change
* Read delta notation for absolute change
||
* Develop discrete models of natural phenomena and use the models to predict future values
* Calculate the carrying capacity and logistic growth rate of a real-world scenario
|-
| Week 13 || Lesson 11.C || [[Logistic Growth Model]] || Determine the increasing and/or decreasing behavior of outputs in a table || Explore the changes of the values of the parameters of a logistic growth model and describe the effect of those changes on the model
|-
| Week 13 || Lesson 11.D || [[Complex Population Growth and Decay Models]]
||
* Find the constant of proportionality and express variables that are jointly proportional
* Determine how jointly proportional variables affect each other in an abstract model
|| Develop a parameterized time series model with more than two dependent variables in a spreadsheet
|-
| Week 14 || Lesson 11.E || [[Analyzing Complex Population Growth and Decay Models]] || Extract data from an academic article and create models for the data
||
* Determine parameters to match a model’s predictions against historical data
* Create a spreadsheet involving the formulas of the model to predict future behavior
|-
| Week 14 || Lesson 12.A || [[Periodic Function]]
||
* Plot points on coordinate axis
* Analyze the shape of a graph and find 12A coordinate values on a graph
||
* Sketch a graph that depicts a periodic phenomenon
* Identify the period and amplitude of a periodic function
* Compare and contrast the graphs of different periodic models
|-
| Week 15 || Lesson 12.B || [[The Sine Function]]
||
* Plot points on a graph
* Understand that constants (Parameters) in an equation control the relationship between the dependent variable and independent variable
||
* Describe the effect that changing one or more parameters has on the graph of a sine function
* Change the parameters of the sine curve to match given criteria
|}

MAT1043

2022-04-15T16:28:27Z

Khanh: /* Introduction to Mathematics */

==Introduction to Mathematics==
[https://catalog.utsa.edu/search/?P=MAT%201043 MAT 1043. Introduction to Mathematics]. (3-0) 3 Credit Hours. (TCCN = MATH 1332)

Prerequisite: Satisfactory performance on a placement examination. This course is designed primarily for the liberal arts major to satisfy the Core Curriculum mathematics requirement. Topics may include logic; proofs; deductive and inductive reasoning; number theory; fundamentals of statistics; basic statistical graphs; causal connections; financial management; functions; linear graphs and modeling; exponential growth and decay; logarithms; fundamentals of probability; fundamentals of geometry; and basic ideas from trigonometry, calculus, and discrete mathematics. (Formerly MTC 1043. Credit cannot be earned for both MAT 1043 and MTC 1043.) May apply toward the Core Curriculum requirement in Mathematics. Generally offered: Fall, Spring, Summer. Course Fees: LRC1 $12; LRS1 $45; STSI $21.

{| class="wikitable sortable"
|-
! Date !! Sections !! Topics !! Prerequisite Skills !! Student learning outcomes
|-
| Week 1 || Lesson 1.A || [[Data]] || Understand what is a learning community
||
* Collect data from daily life
*Work positively in a group to make a decision
|-
| Week 1 || Lesson 1.B || [[Learning Community]] || || Seek and give help to one another inside and outside of class
|-
| Week 1 || Lesson 1.C || [[First-degree equation involving percentages]] || Determine the original amount, given the percentage that a given number is of the original
||
* Create a first-degree equation involving percentages and solve for the variable
* Apply and justify selection strategies to election results and decisions about other issues
|-
| Week 2 || Lesson 1.D || [[Ranking methods]] || ||
* Earning the most votes may not be sufficient to win an election
* Employ the Borda Count method to determine the winner of an election
* Apply and justify selection strategies to election results
|-
| Week 2 || Lesson 2.A || [[Graphical Display]]
||
* Be able to read and interpret graphical displays
* Know the definition of mean
* Know the definition of median
||
* Analyze a variety of graphical displays and interpret them in context
* Compute the mean of a set of data
* Construct a dot plot or histogram from data
|-
| Week 2 || Lesson 2.C || [[Analyzing graphical displays]]
||
* Compare side-by-side graphical displays
* Be able to read and interpret graphical displays
|| Write a contextual analysis of a graphic display in a formal paper including appropriate mathematical language and explanations
|-
| Week 3 || Lesson 3.A || [[Sampling]] || Students should know the symbols for pop. Mean, sample mean, pop. Standard deviation, sample standard deviation
||
* Explain the difference between populations and samples
* Use the characteristics of a sample to describe the population
* Analyze the conclusions of a study and explain the limitations on inferences made
|-
| Week 3 || Lesson 3.B || [[Mean and Central Limit Theorem]] || Students should know how to determine the mean of a data set || Graph sample means and use the central limit theorem to estimate the population mean
|-
| Week 3 || Lesson 3.C || [[Standard Deviation]] || Students should be able to calculate the size of a portion given the size of the whole and a percentage
||
* Use standard deviation to interpret the spread of a data set
* Calculate the percentage of data in a graph region
|-
| Week 4 || Lesson 4.A || [[Probability]]
|| Students should be able to convert between fractions, decimals, and percentages ||
* Calculate theoretical probability of two or more independent events
* Calculate AND and OR probabilities for independent events
|-
| Week 4 || Lesson 4.B || [[Conditional Probability]] || Students should be able to determine a conditional probability || Calculate conditional probabilities for two or more dependent events
|-
| Week 5 || Lesson 5.A || [[Conversions]]
||
* Be able to calculate unit ratios
* Be able to use ratios to convert units
* Be able to perform dimensional analysis
||
* Recognize when converting units is needed
* Use conversion to make comparisons
|-
| Week 5 || Lesson 5.B || [[Index numbers]]
||
* Be able to convert ratios to decimals and percentages, and divide
* Be able to describe types of averages
||
* Perform calculations involving index numbers
* Make and justify decisions and evaluate claims using index numbers
|-
| Week 5 || Lesson 5.C || [[Weighted averages]] || Should be able to calculate mean
||
* Calculate weighted averages
* Use weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 5.D || [[Expected Value]] || Be able to calculate, percentages, means, and weighted averages
be able to find the mean of a data set
||
* Calculate expected value
* Make predictions about real world scenarios
|-
| Week 6 || Lesson 6.A || [[Weighted moving average graphs]]
||
* Weighted moving average
* Graph points
* Find the mean of a data set
||
* Calculate and compare simple and weighted moving averages
* Analyze graphs of moving average data
* Use simple and moving weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 6.B || [[Weighted moving average graphs continued]]
||
* Weighted moving average
* Find the mean of a data set
* Ue a spreadsheet to do calculations and create graphs
||
* Calculate and compare simple and weighted moving averages
* Write a contextual analysis of a graphical display
* Write a ratio or percentage and explain its meaning within a context
|-
| Week 6 || Lesson 7.A || [[Ratios and percentages]] || Be able to write and simplify fractions, create a pie graph, convert fractions to percentages
||
* Determine percentages based on part-to-whole ratios
* Write a ratio or percentage and explain in meaning within a context
* Read a budget, determine values of line items, and draw conclusions about the overall distributions of funds
|-
| Week 7 || Lesson 7.B || [[Part-to-part ratios & Part-to-whole ratios]] || || Construct a pie graph based on ratios and percentages
|-
| Week 7 || Lesson 7.C || [[Absolute change (additive reasoning) & Relative change (multiplicative reasoning)]] || Students should be able to create a line graph from data
||
* Analyze data in a spreadsheet and graphs
* Develop reasonable hypothesis supported by evidence
* Create line graphs and describe patterns in graphs
* Analyze and describe data using absolute (additive) comparison and relative (multiplicative) reasoning
|-
| Week 7 || Lesson 7.D || [[Adjusting claims and hypothesis]] || Students should be able to create a line graph from data || Analyze data in a spreadsheet and graphs to compare changes in categories
|-
| Week 8 || Lesson 7.E || [[Debt-to-income (DTI) ratios]] || Be able to write ratios and proportions, solve proportions, calculate percentages from ratios
||
* Calculate a DTI ratio
* Draw conclusions from DTI about the appropriateness of the percentage of income spent on housing and debt
|-
| Week 8 || Lesson 7.F || [[Proportional reasoning]] || Write rates, convert ratios to percentages
||
* Interpret ratios and percentages as rates of change
* Compare ratios and percentages
* Interpret graphical displays
* Compare mathematical relationships using a variety of representations
|-
| Week 8 || Lesson 8.A || [[Mathematical (Linear) relationships]]
||
* Be able to plot ordered pairs, sketch graphs of linear equations, construct a linear equation given the slope and y intercept
* Write a linear equation based on a verbal description/or that passes through 2 points
* Solve a 2-step linear equation
||
* Compare mathematical relationships using a variety of representations
* Write a linear equation given a slope and y intercept
|-
| Week 9 || Lesson 8.B || [[Proportionality vs. Linearity]] || Determine when 2 quantities are proportional || Explain, compare, and contrast linear and proportional relationships
|-
| Week 9 || Lesson 8.C || [[Simple and Compound Interest (Linear and Exponential Models)]]
||
* Write a percentage as a decimal
* Perform calculations with percentages
||
* Describe the difference between simple and compound interest in practical and mathematical terms
* Compare and contrast patterns in linear and exponential models
|-
| Week 10 || Lesson 8.D || [[Regression]]
||
* Read a scatterplot
* Interpret the slope of a line in context
||
* Create a scatterplot and regression equation using technology and estimate the parameters of the line of best fit
* Interpret the parameters (slope, y-intercept, coefficient of determination) of a simple linear regression
|-
| Week 10 || Lesson 8.E || [[Piecewise Linear Function]]
||
* Use a percentage rate to calculate tax
* Use constant rates of change to write a linear equation
* Identify the slope of a linear function
* Solve a linear equation for a given output
||
* Model a progressive income tax system algebraically and graphically
* Compare a progressive income tax system to a flat tax system and explain advantages and disadvantages of different income tax systems
|-
| Week 10 || Lesson 9.A || [[Depreciation]]
||
* Graph ordered pairs
* Identify the value of the output variable, given the input value, using a graphical and symbolic representation of the relationship between two variables
||
* Interpolate and extrapolate using a graphical representation of the relationship between two variables
* Use a symbolic model to find the exact value of one variable, given the value of the other variable, and relate those values to the context of the problem
|-
| Week 10 || Lesson 9.B || [[Geometric interpretation of interpolation ]]
||
* Graph data from a scenario
* Create proportions using the sides of similar triangles and solve them
||
* Create a proportion between corresponding sides of similar triangles
* Use variables with subscripts
|-
| Week 11 || Lesson 9.C || [[Univariate, Bivariate, Correlation and Causation]]
||
* Calculate measures of central tendency
* Analyze data and visual displays of univariate and bivariate data and describe trends
||
* Create a line graph for univariate data
* Determine, informally, the correlation between bivariate data
* Analyze data and related graphs and describe the trend of the data
|-
| Week 11 || Lesson 9.D || [[Univariate, Bivariate, Correlation and Causation]] || Create and interpret a scatterplot || Explain why, even if there is a strong correlation, a change in one variable may not cause a change in the other
|-
| Week 11 || Lesson 10.A || [[Time series model of exponential growth]] || Use formulas in spreadsheets
||
* Develop a time series model for the Fibonacci problem
* Test whether data are exponential by comparing the rate of growth to the population size
|-
| Week 12 || Lesson 10.B || [[Linear and Exponential Models]]
||
* Create a table of values and scatterplot in a spreadsheet
* Calculate the average rate of growth for a period of time
* Calculate the first differences in a data set
||
* Evaluate the mathematical appropriateness of a model give historical data
* Determine whether a data set suggests a linear or exponential relationship
* Identify and use an appropriate model to predict a future outcome
|-
| Week 12 || Lesson 11.A || [[Continuous Growth]]
||
* Analyze the relationship of input and output values in a problem situation
* Determine whether numerical and graphical relationships are increasing at a constant rate, at an increasing rate or at a decreasing rate
||
* Sketch a model for a population that increases at an increasing rate and a decreasing rate
* Identify behavior in a graph, draw conclusions about the behavior, and predict future outcomes
|-
| Week 12 || Lesson 11.B || [[Carrying Capacity and Logistic Growth Rate]]
||
* Calculate absolute and relative change
* Read delta notation for absolute change
||
* Develop discrete models of natural phenomena and use the models to predict future values
* Calculate the carrying capacity and logistic growth rate of a real-world scenario
|-
| Week 13 || Lesson 11.C || [[Logistic Growth Model]] || Determine the increasing and/or decreasing behavior of outputs in a table || Explore the changes of the values of the parameters of a logistic growth model and describe the effect of those changes on the model
|-
| Week 13 || Lesson 11.D || [[Complex Population Growth and Decay Models]]
||
* Find the constant of proportionality and express variables that are jointly proportional
* Determine how jointly proportional variables affect each other in an abstract model
|| Develop a parameterized time series model with more than two dependent variables in a spreadsheet
|-
| Week 14 || Lesson 11.E || [[Analyzing Complex Population Growth and Decay Models]] || Extract data from an academic article and create models for the data
||
* Determine parameters to match a model’s predictions against historical data
* Create a spreadsheet involving the formulas of the model to predict future behavior
|-
| Week 14 || Lesson 12.A || [[Periodic Function]]
||
* Plot points on coordinate axis
* Analyze the shape of a graph and find 12A coordinate values on a graph
||
* Sketch a graph that depicts a periodic phenomenon
* Identify the period and amplitude of a periodic function
* Compare and contrast the graphs of different periodic models
|-
| Week 15 || Lesson 12.B || [[The Sine Function]]
||
* Plot points on a graph
* Understand that constants (Parameters) in an equation control the relationship between the dependent variable and independent variable
||
* Describe the effect that changing one or more parameters has on the graph of a sine function
* Change the parameters of the sine curve to match given criteria
|}

MAT1043

2022-04-15T16:18:12Z

Khanh: /* Introduction to Mathematics */

==Introduction to Mathematics==
[https://catalog.utsa.edu/search/?P=MAT%201043 MAT 1043. Introduction to Mathematics]. (3-0) 3 Credit Hours. (TCCN = MATH 1332)

Prerequisite: Satisfactory performance on a placement examination. This course is designed primarily for the liberal arts major to satisfy the Core Curriculum mathematics requirement. Topics may include logic; proofs; deductive and inductive reasoning; number theory; fundamentals of statistics; basic statistical graphs; causal connections; financial management; functions; linear graphs and modeling; exponential growth and decay; logarithms; fundamentals of probability; fundamentals of geometry; and basic ideas from trigonometry, calculus, and discrete mathematics. (Formerly MTC 1043. Credit cannot be earned for both MAT 1043 and MTC 1043.) May apply toward the Core Curriculum requirement in Mathematics. Generally offered: Fall, Spring, Summer. Course Fees: LRC1 $12; LRS1 $45; STSI $21.

{| class="wikitable sortable"
|-
! Date !! Sections !! Topics !! Prerequisite Skills !! Student learning outcomes
|-
| Week 1 || Lesson 1.A || [[Data]] || Understand what is a learning community
||
* Collect data from daily life
*Work positively in a group to make a decision
|-
| Week 1 || Lesson 1.B || [[Learning Community]] || || Seek and give help to one another inside and outside of class
|-
| Week 1 || Lesson 1.C || [[First-degree equation involving percentages]] || Determine the original amount, given the percentage that a given number is of the original
||
* Create a first-degree equation involving percentages and solve for the variable
* Apply and justify selection strategies to election results and decisions about other issues
|-
| Week 2 || Lesson 1.D || [[Ranking methods]] || ||
|-
| Week 2 || Lesson 2.A || [[Graphical Display]]
||
* Be able to read and interpret graphical displays
* Know the definition of mean
* Know the definition of median
||
* Analyze a variety of graphical displays and interpret them in context
* Compute the mean of a set of data
* Construct a dot plot or histogram from data
|-
| Week 2 || Lesson 2.C || [[Analyzing graphical displays]]
||
* Compare side-by-side graphical displays
* Be able to read and interpret graphical displays
|| Write a contextual analysis of a graphic display in a formal paper including appropriate mathematical language and explanations
|-
| Week 3 || Lesson 3.A || [[Sampling]] || Students should know the symbols for pop. Mean, sample mean, pop. Standard deviation, sample standard deviation
||
* Explain the difference between populations and samples
* Use the characteristics of a sample to describe the population
* Analyze the conclusions of a study and explain the limitations on inferences made
|-
| Week 3 || Lesson 3.B || [[Mean and Central Limit Theorem]] || Students should know how to determine the mean of a data set || Graph sample means and use the central limit theorem to estimate the population mean
|-
| Week 3 || Lesson 3.C || [[Standard Deviation]] || Students should be able to calculate the size of a portion given the size of the whole and a percentage
||
* Use standard deviation to interpret the spread of a data set
* Calculate the percentage of data in a graph region
|-
| Week 4 || Lesson 4.A || [[Probability]]
|| Students should be able to convert between fractions, decimals, and percentages ||
* Calculate theoretical probability of two or more independent events
* Calculate AND and OR probabilities for independent events
|-
| Week 4 || Lesson 4.B || [[Conditional Probability]] || Students should be able to determine a conditional probability || Calculate conditional probabilities for two or more dependent events
|-
| Week 5 || Lesson 5.A || [[Conversions]]
||
* Be able to calculate unit ratios
* Be able to use ratios to convert units
* Be able to perform dimensional analysis
||
* Recognize when converting units is needed
* Use conversion to make comparisons
|-
| Week 5 || Lesson 5.B || [[Index numbers]]
||
* Be able to convert ratios to decimals and percentages, and divide
* Be able to describe types of averages
||
* Perform calculations involving index numbers
* Make and justify decisions and evaluate claims using index numbers
|-
| Week 5 || Lesson 5.C || [[Weighted averages]] || Should be able to calculate mean
||
* Calculate weighted averages
* Use weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 5.D || [[Expected Value]] || Be able to calculate, percentages, means, and weighted averages
be able to find the mean of a data set
||
* Calculate expected value
* Make predictions about real world scenarios
|-
| Week 6 || Lesson 6.A || [[Weighted moving average graphs]]
||
* Weighted moving average
* Graph points
* Find the mean of a data set
||
* Calculate and compare simple and weighted moving averages
* Analyze graphs of moving average data
* Use simple and moving weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 6.B || [[Weighted moving average graphs continued]]
||
* Weighted moving average
* Find the mean of a data set
* Ue a spreadsheet to do calculations and create graphs
||
* Calculate and compare simple and weighted moving averages
* Write a contextual analysis of a graphical display
* Write a ratio or percentage and explain its meaning within a context
|-
| Week 6 || Lesson 7.A || [[Ratios and percentages]] || Be able to write and simplify fractions, create a pie graph, convert fractions to percentages
||
* Determine percentages based on part-to-whole ratios
* Write a ratio or percentage and explain in meaning within a context
* Read a budget, determine values of line items, and draw conclusions about the overall distributions of funds
|-
| Week 7 || Lesson 7.B || [[Part-to-part ratios & Part-to-whole ratios]] || || Construct a pie graph based on ratios and percentages
|-
| Week 7 || Lesson 7.C || [[Absolute change (additive reasoning) & Relative change (multiplicative reasoning)]] || Students should be able to create a line graph from data
||
* Analyze data in a spreadsheet and graphs
* Develop reasonable hypothesis supported by evidence
* Create line graphs and describe patterns in graphs
* Analyze and describe data using absolute (additive) comparison and relative (multiplicative) reasoning
|-
| Week 7 || Lesson 7.D || [[Adjusting claims and hypothesis]] || Students should be able to create a line graph from data || Analyze data in a spreadsheet and graphs to compare changes in categories
|-
| Week 8 || Lesson 7.E || [[Debt-to-income (DTI) ratios]] || Be able to write ratios and proportions, solve proportions, calculate percentages from ratios
||
* Calculate a DTI ratio
* Draw conclusions from DTI about the appropriateness of the percentage of income spent on housing and debt
|-
| Week 8 || Lesson 7.F || [[Proportional reasoning]] || Write rates, convert ratios to percentages
||
* Interpret ratios and percentages as rates of change
* Compare ratios and percentages
* Interpret graphical displays
* Compare mathematical relationships using a variety of representations
|-
| Week 8 || Lesson 8.A || [[Mathematical (Linear) relationships]]
||
* Be able to plot ordered pairs, sketch graphs of linear equations, construct a linear equation given the slope and y intercept
* Write a linear equation based on a verbal description/or that passes through 2 points
* Solve a 2-step linear equation
||
* Compare mathematical relationships using a variety of representations
* Write a linear equation given a slope and y intercept
|-
| Week 9 || Lesson 8.B || [[Proportionality vs. Linearity]] || Determine when 2 quantities are proportional || Explain, compare, and contrast linear and proportional relationships
|-
| Week 9 || Lesson 8.C || [[Simple and Compound Interest (Linear and Exponential Models)]]
||
* Write a percentage as a decimal
* Perform calculations with percentages
||
* Describe the difference between simple and compound interest in practical and mathematical terms
* Compare and contrast patterns in linear and exponential models
|-
| Week 10 || Lesson 8.D || [[Regression]]
||
* Read a scatterplot
* Interpret the slope of a line in context
||
* Create a scatterplot and regression equation using technology and estimate the parameters of the line of best fit
* Interpret the parameters (slope, y-intercept, coefficient of determination) of a simple linear regression
|-
| Week 10 || Lesson 8.E || [[Piecewise Linear Function]]
||
* Use a percentage rate to calculate tax
* Use constant rates of change to write a linear equation
* Identify the slope of a linear function
* Solve a linear equation for a given output
||
* Model a progressive income tax system algebraically and graphically
* Compare a progressive income tax system to a flat tax system and explain advantages and disadvantages of different income tax systems
|-
| Week 10 || Lesson 9.A || [[Depreciation]]
||
* Graph ordered pairs
* Identify the value of the output variable, given the input value, using a graphical and symbolic representation of the relationship between two variables
||
* Interpolate and extrapolate using a graphical representation of the relationship between two variables
* Use a symbolic model to find the exact value of one variable, given the value of the other variable, and relate those values to the context of the problem
|-
| Week 10 || Lesson 9.B || [[Geometric interpretation of interpolation ]]
||
* Graph data from a scenario
* Create proportions using the sides of similar triangles and solve them
||
* Create a proportion between corresponding sides of similar triangles
* Use variables with subscripts
|-
| Week 11 || Lesson 9.C || [[Univariate, Bivariate, Correlation and Causation]]
||
* Calculate measures of central tendency
* Analyze data and visual displays of univariate and bivariate data and describe trends
||
* Create a line graph for univariate data
* Determine, informally, the correlation between bivariate data
* Analyze data and related graphs and describe the trend of the data
|-
| Week 11 || Lesson 9.D || [[Univariate, Bivariate, Correlation and Causation]] || Create and interpret a scatterplot || Explain why, even if there is a strong correlation, a change in one variable may not cause a change in the other
|-
| Week 11 || Lesson 10.A || [[Time series model of exponential growth]] || Use formulas in spreadsheets
||
* Develop a time series model for the Fibonacci problem
* Test whether data are exponential by comparing the rate of growth to the population size
|-
| Week 12 || Lesson 10.B || [[Linear and Exponential Models]]
||
* Create a table of values and scatterplot in a spreadsheet
* Calculate the average rate of growth for a period of time
* Calculate the first differences in a data set
||
* Evaluate the mathematical appropriateness of a model give historical data
* Determine whether a data set suggests a linear or exponential relationship
* Identify and use an appropriate model to predict a future outcome
|-
| Week 12 || Lesson 11.A || [[Continuous Growth]]
||
* Analyze the relationship of input and output values in a problem situation
* Determine whether numerical and graphical relationships are increasing at a constant rate, at an increasing rate or at a decreasing rate
||
* Sketch a model for a population that increases at an increasing rate and a decreasing rate
* Identify behavior in a graph, draw conclusions about the behavior, and predict future outcomes
|-
| Week 12 || Lesson 11.B || [[Carrying Capacity and Logistic Growth Rate]]
||
* Calculate absolute and relative change
* Read delta notation for absolute change
||
* Develop discrete models of natural phenomena and use the models to predict future values
* Calculate the carrying capacity and logistic growth rate of a real-world scenario
|-
| Week 13 || Lesson 11.C || [[Logistic Growth Model]] || Determine the increasing and/or decreasing behavior of outputs in a table || Explore the changes of the values of the parameters of a logistic growth model and describe the effect of those changes on the model
|-
| Week 13 || Lesson 11.D || [[Complex Population Growth and Decay Models]]
||
* Find the constant of proportionality and express variables that are jointly proportional
* Determine how jointly proportional variables affect each other in an abstract model
|| Develop a parameterized time series model with more than two dependent variables in a spreadsheet
|-
| Week 14 || Lesson 11.E || [[Analyzing Complex Population Growth and Decay Models]] || Extract data from an academic article and create models for the data
||
* Determine parameters to match a model’s predictions against historical data
* Create a spreadsheet involving the formulas of the model to predict future behavior
|-
| Week 14 || Lesson 12.A || [[Periodic Function]]
||
* Plot points on coordinate axis
* Analyze the shape of a graph and find 12A coordinate values on a graph
||
* Sketch a graph that depicts a periodic phenomenon
* Identify the period and amplitude of a periodic function
* Compare and contrast the graphs of different periodic models
|-
| Week 15 || Lesson 12.B || [[The Sine Function]]
||
* Plot points on a graph
* Understand that constants (Parameters) in an equation control the relationship between the dependent variable and independent variable
||
* Describe the effect that changing one or more parameters has on the graph of a sine function
* Change the parameters of the sine curve to match given criteria
|}

MAT1043

2022-04-15T16:15:46Z

Khanh: /* Introduction to Mathematics */

==Introduction to Mathematics==
[https://catalog.utsa.edu/search/?P=MAT%201043 MAT 1043. Introduction to Mathematics]. (3-0) 3 Credit Hours. (TCCN = MATH 1332)

Prerequisite: Satisfactory performance on a placement examination. This course is designed primarily for the liberal arts major to satisfy the Core Curriculum mathematics requirement. Topics may include logic; proofs; deductive and inductive reasoning; number theory; fundamentals of statistics; basic statistical graphs; causal connections; financial management; functions; linear graphs and modeling; exponential growth and decay; logarithms; fundamentals of probability; fundamentals of geometry; and basic ideas from trigonometry, calculus, and discrete mathematics. (Formerly MTC 1043. Credit cannot be earned for both MAT 1043 and MTC 1043.) May apply toward the Core Curriculum requirement in Mathematics. Generally offered: Fall, Spring, Summer. Course Fees: LRC1 $12; LRS1 $45; STSI $21.

{| class="wikitable sortable"
|-
! Date !! Sections !! Topics !! Prerequisite Skills !! Student learning outcomes
|-
| Week 1 || Lesson 1.A || [[Data]] || Understand what is a learning community
||
* Collect data from daily life
*Work positively in a group to make a decision
|-
| Week 1 || Lesson 1.B || [[Learning Community]] || || Seek and give help to one another inside and outside of class
|-
| Week 1 || Lesson 1.C || [[First-degree equation involving percentages]] || Determine the original amount, given the percentage that a given number is of the original
||
* Create a first-degree equation involving percentages and solve for the variable
* Apply and justify selection strategies to election results and decisions about other issues
|-
| Week 2 || Lesson 1.D || [[Ranking methods]] || ||
|-
| Week 2 || Lesson 2.A || [[Graphical Display]]
||
* Be able to read and interpret graphical displays
* Know the definition of mean
* Know the definition of median
||
* Analyze a variety of graphical displays and interpret them in context
* Compute the mean of a set of data
* Construct a dot plot or histogram from data
|-
| Week 2 || Lesson 2.C || [[Analyzing graphical displays]]
||
* Compare side-by-side graphical displays
* Be able to read and interpret graphical displays
|| Write a contextual analysis of a graphic display in a formal paper including appropriate mathematical language and explanations
|-
| Week 3 || Lesson 3.A || [[Sampling]] || Students should know the symbols for pop. Mean, sample mean, pop. Standard deviation, sample standard deviation
||
* Explain the difference between populations and samples
* Use the characteristics of a sample to describe the population
* Analyze the conclusions of a study and explain the limitations on inferences made
|-
| Week 3 || Lesson 3.B || [[Mean and Central Limit Theorem]] || Students should know how to determine the mean of a data set || Graph sample means and use the central limit theorem to estimate the population mean
|-
| Week 3 || Lesson 3.C || [[Standard Deviation]] || Students should be able to calculate the size of a portion given the size of the whole and a percentage
||
* Use standard deviation to interpret the spread of a data set
* Calculate the percentage of data in a graph region
|-
| Week 4 || Lesson 4.A || [[Probability]]
|| Students should be able to convert between fractions, decimals, and percentages ||
* Calculate theoretical probability of two or more independent events
* Calculate AND and OR probabilities for independent events
|-
| Week 4 || Lesson 4.B || [[Conditional Probability]] || Students should be able to determine a conditional probability || Calculate conditional probabilities for two or more dependent events
|-
| Week 5 || Lesson 5.A || [[Conversions]]
||
* Be able to calculate unit ratios
* Be able to use ratios to convert units
* Be able to perform dimensional analysis
||
* Recognize when converting units is needed
* Use conversion to make comparisons
|-
| Week 5 || Lesson 5.B || [[Index numbers]]
||
* Be able to convert ratios to decimals and percentages, and divide
* Be able to describe types of averages
||
* Perform calculations involving index numbers
* Make and justify decisions and evaluate claims using index numbers
|-
| Week 5 || Lesson 5.C || [[Weighted averages]] || Should be able to calculate mean
||
* Calculate weighted averages
* Use weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 5.D || [[Expected Value]] || Be able to calculate, percentages, means, and weighted averages
be able to find the mean of a data set
||
* Calculate expected value
* Make predictions about real world scenarios
|-
| Week 6 || Lesson 6.A || [[Weighted moving average graphs]]
||
* Weighted moving average
* Graph points
* Find the mean of a data set
||
* Calculate and compare simple and weighted moving averages
* Analyze graphs of moving average data
* Use simple and moving weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 6.B || [[Weighted moving average graphs continued]]
||
* Weighted moving average
* Find the mean of a data set
* Ue a spreadsheet to do calculations and create graphs
||
* Calculate and compare simple and weighted moving averages
* Write a contextual analysis of a graphical display
* Write a ratio or percentage and explain its meaning within a context
|-
| Week 6 || Lesson 7.A || [[Ratios and percentages]] || Be able to write and simplify fractions, create a pie graph, convert fractions to percentages
||
* Determine percentages based on part-to-whole ratios
* Write a ratio or percentage and explain in meaning within a context
* Read a budget, determine values of line items, and draw conclusions about the overall distributions of funds
|-
| Week 7 || Lesson 7.B || [[Part-to-part ratios & Part-to-whole ratios]] || || Construct a pie graph based on ratios and percentages
|-
| Week 7 || Lesson 7.C || [[Absolute change (additive reasoning) & Relative change (multiplicative reasoning)]] || Students should be able to create a line graph from data
||
* Analyze data in a spreadsheet and graphs
* Develop reasonable hypothesis supported by evidence
* Create line graphs and describe patterns in graphs
|-
| Week 7 || Lesson 7.D || [[Adjusting claims and hypothesis]] || Students should be able to create a line graph from data || Analyze data in a spreadsheet and graphs to compare changes in categories
|-
| Week 8 || Lesson 7.E || [[Debt-to-income (DTI) ratios]] || Be able to write ratios and proportions, solve proportions, calculate percentages from ratios
||
* Calculate a DTI ratio
* Draw conclusions from DTI about the appropriateness of the percentage of income spent on housing and debt
|-
| Week 8 || Lesson 7.F || [[Proportional reasoning]] || Write rates, convert ratios to percentages
||
* Interpret ratios and percentages as rates of change
* Compare ratios and percentages
* Interpret graphical displays
* Compare mathematical relationships using a variety of representations
|-
| Week 8 || Lesson 8.A || [[Mathematical (Linear) relationships]]
||
* Be able to plot ordered pairs, sketch graphs of linear equations, construct a linear equation given the slope and y intercept
* Write a linear equation based on a verbal description/or that passes through 2 points
* Solve a 2-step linear equation
||
* Compare mathematical relationships using a variety of representations
* Write a linear equation given a slope and y intercept
|-
| Week 9 || Lesson 8.B || [[Proportionality vs. Linearity]] || Determine when 2 quantities are proportional || Explain, compare, and contrast linear and proportional relationships
|-
| Week 9 || Lesson 8.C || [[Simple and Compound Interest (Linear and Exponential Models)]]
||
* Write a percentage as a decimal
* Perform calculations with percentages
||
* Describe the difference between simple and compound interest in practical and mathematical terms
* Compare and contrast patterns in linear and exponential models
|-
| Week 10 || Lesson 8.D || [[Regression]]
||
* Read a scatterplot
* Interpret the slope of a line in context
||
* Create a scatterplot and regression equation using technology and estimate the parameters of the line of best fit
* Interpret the parameters (slope, y-intercept, coefficient of determination) of a simple linear regression
|-
| Week 10 || Lesson 8.E || [[Piecewise Linear Function]]
||
* Use a percentage rate to calculate tax
* Use constant rates of change to write a linear equation
* Identify the slope of a linear function
* Solve a linear equation for a given output
||
* Model a progressive income tax system algebraically and graphically
* Compare a progressive income tax system to a flat tax system and explain advantages and disadvantages of different income tax systems
|-
| Week 10 || Lesson 9.A || [[Depreciation]]
||
* Graph ordered pairs
* Identify the value of the output variable, given the input value, using a graphical and symbolic representation of the relationship between two variables
||
* Interpolate and extrapolate using a graphical representation of the relationship between two variables
* Use a symbolic model to find the exact value of one variable, given the value of the other variable, and relate those values to the context of the problem
|-
| Week 10 || Lesson 9.B || [[Geometric interpretation of interpolation ]]
||
* Graph data from a scenario
* Create proportions using the sides of similar triangles and solve them
||
* Create a proportion between corresponding sides of similar triangles
* Use variables with subscripts
|-
| Week 11 || Lesson 9.C || [[Univariate, Bivariate, Correlation and Causation]]
||
* Calculate measures of central tendency
* Analyze data and visual displays of univariate and bivariate data and describe trends
||
* Create a line graph for univariate data
* Determine, informally, the correlation between bivariate data
* Analyze data and related graphs and describe the trend of the data
|-
| Week 11 || Lesson 9.D || [[Univariate, Bivariate, Correlation and Causation]] || Create and interpret a scatterplot || Explain why, even if there is a strong correlation, a change in one variable may not cause a change in the other
|-
| Week 11 || Lesson 10.A || [[Time series model of exponential growth]] || Use formulas in spreadsheets
||
* Develop a time series model for the Fibonacci problem
* Test whether data are exponential by comparing the rate of growth to the population size
|-
| Week 12 || Lesson 10.B || [[Linear and Exponential Models]]
||
* Create a table of values and scatterplot in a spreadsheet
* Calculate the average rate of growth for a period of time
* Calculate the first differences in a data set
||
* Evaluate the mathematical appropriateness of a model give historical data
* Determine whether a data set suggests a linear or exponential relationship
* Identify and use an appropriate model to predict a future outcome
|-
| Week 12 || Lesson 11.A || [[Continuous Growth]]
||
* Analyze the relationship of input and output values in a problem situation
* Determine whether numerical and graphical relationships are increasing at a constant rate, at an increasing rate or at a decreasing rate
||
* Sketch a model for a population that increases at an increasing rate and a decreasing rate
* Identify behavior in a graph, draw conclusions about the behavior, and predict future outcomes
|-
| Week 12 || Lesson 11.B || [[Carrying Capacity and Logistic Growth Rate]]
||
* Calculate absolute and relative change
* Read delta notation for absolute change
||
* Develop discrete models of natural phenomena and use the models to predict future values
* Calculate the carrying capacity and logistic growth rate of a real-world scenario
|-
| Week 13 || Lesson 11.C || [[Logistic Growth Model]] || Determine the increasing and/or decreasing behavior of outputs in a table || Explore the changes of the values of the parameters of a logistic growth model and describe the effect of those changes on the model
|-
| Week 13 || Lesson 11.D || [[Complex Population Growth and Decay Models]]
||
* Find the constant of proportionality and express variables that are jointly proportional
* Determine how jointly proportional variables affect each other in an abstract model
|| Develop a parameterized time series model with more than two dependent variables in a spreadsheet
|-
| Week 14 || Lesson 11.E || [[Analyzing Complex Population Growth and Decay Models]] || Extract data from an academic article and create models for the data
||
* Determine parameters to match a model’s predictions against historical data
* Create a spreadsheet involving the formulas of the model to predict future behavior
|-
| Week 14 || Lesson 12.A || [[Periodic Function]]
||
* Plot points on coordinate axis
* Analyze the shape of a graph and find 12A coordinate values on a graph
||
* Sketch a graph that depicts a periodic phenomenon
* Identify the period and amplitude of a periodic function
* Compare and contrast the graphs of different periodic models
|-
| Week 15 || Lesson 12.B || [[The Sine Function]]
||
* Plot points on a graph
* Understand that constants (Parameters) in an equation control the relationship between the dependent variable and independent variable
||
* Describe the effect that changing one or more parameters has on the graph of a sine function
* Change the parameters of the sine curve to match given criteria
|}

MAT1043

2022-04-15T16:13:30Z

Khanh:

==Introduction to Mathematics==
[https://catalog.utsa.edu/search/?P=MAT%201043 MAT 1043. Introduction to Mathematics]. (3-0) 3 Credit Hours. (TCCN = MATH 1332)

Prerequisite: Satisfactory performance on a placement examination. This course is designed primarily for the liberal arts major to satisfy the Core Curriculum mathematics requirement. Topics may include logic; proofs; deductive and inductive reasoning; number theory; fundamentals of statistics; basic statistical graphs; causal connections; financial management; functions; linear graphs and modeling; exponential growth and decay; logarithms; fundamentals of probability; fundamentals of geometry; and basic ideas from trigonometry, calculus, and discrete mathematics. (Formerly MTC 1043. Credit cannot be earned for both MAT 1043 and MTC 1043.) May apply toward the Core Curriculum requirement in Mathematics. Generally offered: Fall, Spring, Summer. Course Fees: LRC1 $12; LRS1 $45; STSI $21.

{| class="wikitable sortable"
|-
! Date !! Sections !! Topics !! Prerequisite Skills !! Student learning outcomes
|-
| Week 1 || Lesson 1.A || [[Data]] || Understand what is a learning community
||
* Collect data from daily life
*Work positively in a group to make a decision
|-
| Week 1 || Lesson 1.B || [[Learning Community]] || || Seek and give help to one another inside and outside of class
|-
| Week 1 || Lesson 1.C || [[First-degree equation involving percentages]] || Determine the original amount, given the percentage that a given number is of the original
||
* Create a first-degree equation involving percentages and solve for the variable
* Apply and justify selection strategies to election results and decisions about other issues
|-
| Week 2 || Lesson 1.D || [[Ranking methods]] || ||
|-
| Week 2 || Lesson 2.A || [[Graphical Display]]
||
* Be able to read and interpret graphical displays
* Know the definition of mean
* Know the definition of median
||
* Analyze a variety of graphical displays and interpret them in context
* Compute the mean of a set of data
* Construct a dot plot or histogram from data
|-
| Week 2 || Lesson 2.C || [[Analyzing graphical displays]]
||
* Compare side-by-side graphical displays
* Be able to read and interpret graphical displays
|| Write a contextual analysis of a graphic display in a formal paper including appropriate mathematical language and explanations
|-
| Week 3 || Lesson 3.A || [[Sampling]] || Students should know the symbols for pop. Mean, sample mean, pop. Standard deviation, sample standard deviation
||
* Explain the difference between populations and samples
* Use the characteristics of a sample to describe the population
* Analyze the conclusions of a study and explain the limitations on inferences made
|-
| Week 3 || Lesson 3.B || [[Mean and Central Limit Theorem]] || Students should know how to determine the mean of a data set || Graph sample means and use the central limit theorem to estimate the population mean
|-
| Week 3 || Lesson 3.C || [[Standard Deviation]] || Students should be able to calculate the size of a portion given the size of the whole and a percentage
||
* Use standard deviation to interpret the spread of a data set
* Calculate the percentage of data in a graph region
|-
| Week 4 || Lesson 4.A || [[Probability]]
|| Students should be able to convert between fractions, decimals, and percentages ||
* Calculate theoretical probability of two or more independent events
* Calculate AND and OR probabilities for independent events
|-
| Week 4 || Lesson 4.B || [[Conditional Probability]] || Students should be able to determine a conditional probability || Calculate conditional probabilities for two or more dependent events
|-
| Week 5 || Lesson 5.A || [[Conversions]]
||
* Be able to calculate unit ratios
* Be able to use ratios to convert units
* Be able to perform dimensional analysis
||
* Recognize when converting units is needed
* Use conversion to make comparisons
|-
| Week 5 || Lesson 5.B || [[Index numbers]]
||
* Be able to convert ratios to decimals and percentages, and divide
* Be able to describe types of averages
||
* Perform calculations involving index numbers
* Make and justify decisions and evaluate claims using index numbers
|-
| Week 5 || Lesson 5.C || [[Weighted averages]] || Should be able to calculate mean
||
* Calculate weighted averages
* Use weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 5.D || [[Expected Value]] || Be able to calculate, percentages, means, and weighted averages
be able to find the mean of a data set
||
* Calculate expected value
* Make predictions about real world scenarios
|-
| Week 6 || Lesson 6.A || [[Weighted moving average graphs]]
||
* Weighted moving average
* Graph points
* Find the mean of a data set
||
* Calculate and compare simple and weighted moving averages
* Analyze graphs of moving average data
* Use simple and moving weighted averages to analyze data and draw conclusions about the data
|-
| Week 6 || Lesson 6.B || [[Weighted moving average graphs continued]]
||
* Weighted moving average
* Find the mean of a data set
* Ue a spreadsheet to do calculations and create graphs
||
* Calculate and compare simple and weighted moving averages
* Write a contextual analysis of a graphical display
* Write a ratio or percentage and explain its meaning within a context
|-
| Week 6 || Lesson 7.A || [[Ratios and percentages]] || Be able to write and simplify fractions, create a pie graph, convert fractions to percentages
||
* Determine percentages based on part-to-whole ratios
* Write a ratio or percentage and explain in meaning within a context
* Read a budget, determine values of line items, and draw conclusions about the overall distributions of funds
|-
| Week 7 || Lesson 7.B || [[Part-to-part ratios & Part-to-whole ratios]] || || Construct a pie graph based on ratios and percentages
|-
| Week 7 || Lesson 7.C || [[Absolute change (additive reasoning) & Relative change (multiplicative reasoning)]] || Students should be able to create a line graph from data
||
* Analyze data in a spreadsheet and graphs
* Develop reasonable hypothesis supported by evidence
* Create line graphs and describe patterns in graphs
|-
| Week 7 || Lesson 7.D || [[Adjusting claims and hypothesis]] || Students should be able to create a line graph from data || Analyze data in a spreadsheet and graphs to compare changes in categories
|-
| Week 8 || Lesson 7.E || [[Debt-to-income (DTI) ratios]] || Be able to write ratios and proportions, solve proportions, calculate percentages from ratios
||
* Calculate a DTI ratio
* Draw conclusions from DTI about the appropriateness of the percentage of income spent on housing and debt
|-
| Week 8 || Lesson 7.F || [[Proportional reasoning]] || Write rates, convert ratios to percentages
||
* Interpret ratios and percentages as rates of change
* Compare ratios and percentages
* Interpret graphical displays
* Compare mathematical relationships using a variety of representations
|-
| Week 8 || Lesson 8.A || [[Mathematical (Linear) relationships]]
||
* Be able to plot ordered pairs, sketch graphs of linear equations, construct a linear equation given the slope and y intercept
* Write a linear equation based on a verbal description/or that passes through 2 points
* Solve a 2-step linear equation
||
* Compare mathematical relationships using a variety of representations
* Write a linear equation given a slope and y intercept
|-
| Week 9 || Lesson 8.B || [[Proportionality vs. Linearity]] || Determine when 2 quantities are proportional || Explain, compare, and contrast linear and proportional relationships
|-
| Week 9 || Lesson 8.C || [[Simple and Compound Interest (Linear and Exponential Models)]]
||
* Write a percentage as a decimal
* Perform calculations with percentages
||
* Describe the difference between simple and compound interest in practical and mathematical terms
* Compare and contrast patterns in linear and exponential models
|-
| Week 10 || Lesson 8.D || [[Regression]]
||
* Read a scatterplot
* Interpret the slope of a line in context
||
* Create a scatterplot and regression equation using technology and estimate the parameters of the line of best fit
* Interpret the parameters (slope, y-intercept, coefficient of determination) of a simple linear regression
|-
| Week 10 || Lesson 8.E || [[Piecewise Linear Function]]
||
* Use a percentage rate to calculate tax
* Use constant rates of change to write a linear equation
* Identify the slope of a linear function
* Solve a linear equation for a given output
||
* Model a progressive income tax system algebraically and graphically
* Compare a progressive income tax system to a flat tax system and explain advantages and disadvantages of different income tax systems
|-
| Week 10 || Lesson 9.A || [[Depreciation]]
||
* Graph ordered pairs
* Identify the value of the output variable, given the input value, using a graphical and symbolic representation of the relationship between two variables
||
* Interpolate and extrapolate using a graphical representation of the relationship between two variables
* Use a symbolic model to find the exact value of one variable, given the value of the other variable, and relate those values to the context of the problem
|-
| Week 10 || Lesson 9.B || [[Geometric interpretation of interpolation ]]
||
* Graph data from a scenario
* Create proportions using the sides of similar triangles and solve them
||
* Create a proportion between corresponding sides of similar triangles
* Use variables with subscripts
|-
| Week 11 || Lesson 9.C || [[Univariate, Bivariate, Correlation and Causation]]
||
* Calculate measures of central tendency
* Analyze data and visual displays of univariate and bivariate data and describe trends
||
* Create a line graph for univariate data
* Determine, informally, the correlation between bivariate data
* Analyze data and related graphs and describe the trend of the data
|-
| Week 11 || Lesson 9.D || [[Univariate, Bivariate, Correlation and Causation]] || Create and interpret a scatterplot || Explain why, even if there is a strong correlation, a change in one variable may not cause a change in the other
|-
| Week 11 || Lesson 10.A || [[Time series model of exponential growth]] || Use formulas in spreadsheets
||
* Develop a time series model for the Fibonacci problem
* Test whether data are exponential by comparing the rate of growth to the population size
|-
| Week 12 || Lesson 10.B || [[Linear and Exponential Models]]
||
* Create a table of values and scatterplot in a spreadsheet
* Calculate the average rate of growth for a period of time
* Calculate the first differences in a data set
||
* Evaluate the mathematical appropriateness of a model give historical data
* Determine whether a data set suggests a linear or exponential relationship
* Use an appropriate model to predict a future outcome
|-
| Week 12 || Lesson 11.A || [[Continuous Growth]]
||
* Analyze the relationship of input and output values in a problem situation
* Determine whether numerical and graphical relationships are increasing at a constant rate, at an increasing rate or at a decreasing rate
||
* Sketch a model for a population that increases at an increasing rate and a decreasing rate
* Identify behavior in a graph, draw conclusions about the behavior, and predict future outcomes
|-
| Week 12 || Lesson 11.B || [[Carrying Capacity and Logistic Growth Rate]]
||
* Calculate absolute and relative change
* Read delta notation for absolute change
||
* Develop discrete models of natural phenomena and use the models to predict future values
* Calculate the carrying capacity and logistic growth rate of a real-world scenario
|-
| Week 13 || Lesson 11.C || [[Logistic Growth Model]] || Determine the increasing and/or decreasing behavior of outputs in a table || Explore the changes of the values of the parameters of a logistic growth model and describe the effect of those changes on the model
|-
| Week 13 || Lesson 11.D || [[Complex Population Growth and Decay Models]]
||
* Find the constant of proportionality and express variables that are jointly proportional
* Determine how jointly proportional variables affect each other in an abstract model
|| Develop a parameterized time series model with more than two dependent variables in a spreadsheet
|-
| Week 14 || Lesson 11.E || [[Analyzing Complex Population Growth and Decay Models]] || Extract data from an academic article and create models for the data
||
* Determine parameters to match a model’s predictions against historical data
* Create a spreadsheet involving the formulas of the model to predict future behavior
|-
| Week 14 || Lesson 12.A || [[Periodic Function]]
||
* Plot points on coordinate axis
* Analyze the shape of a graph and find 12A coordinate values on a graph
||
* Sketch a graph that depicts a periodic phenomenon
* Identify the period and amplitude of a periodic function
* Compare and contrast the graphs of different periodic models
|-
| Week 15 || Lesson 12.B || [[The Sine Function]]
||
* Plot points on a graph
* Understand that constants (Parameters) in an equation control the relationship between the dependent variable and independent variable
||
* Describe the effect that changing one or more parameters has on the graph of a sine function
* Change the parameters of the sine curve to match given criteria
|}

Triangle Inequality

2022-02-13T22:35:25Z

Khanh: /* Euclidean geometry */

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, ''π'']) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC </math>, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides <math>\overline{AB} = \overline{AC}</math> divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides <math>\overline{AB} = \overline{AC}</math>. From the triangle postulate, the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But <math>\overline{AB} = \overline{AC}</math>. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows <math> \overline{AC} > \overline{DC}</math>, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the triangle postulate), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that ''d'' = ''a''/3, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation ''r''2 + ''r'' − 1 = 0, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation
''r''2 − ''r'' − 1 = 0, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that <math> r = \sqrt{\phi}</math> it generates a right triangle that is always similar to the Kepler triangle.

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have <math> b^2 = h^2 + d^2 </math> and <math> a^2 = h^2 + (c - d)^2 </math> according to the figure at the right. Subtracting these yields <math> a^2 - b^2 = c^2 - 2cd </math>. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that <math> h^2 = b^2 - d^2 </math>. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., ''x'' = 0 or ''y'' = 0, or
''x'' = ''α y'' for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote ''x'' = (1, 0) and ''y'' = (0, 1). Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, <math> d(x_n, x_m) \leq d(x_n, x) + d(x_m, x) < \epsilon /2 + \epsilon /2 = \epsilon </math>, so that the sequence {{math|{''xn''}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T22:32:32Z

Khanh:

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, ''π'']) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides <math>\overline{AB} = \overline{AC}</math> divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides <math>\overline{AB} = \overline{AC}</math>. From the triangle postulate, the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But <math>\overline{AB} = \overline{AC}</math>. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows <math> \overline{AC} > \overline{DC}</math>, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the triangle postulate), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that ''d'' = ''a''/3, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation ''r''2 + ''r'' − 1 = 0, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation
''r''2 − ''r'' − 1 = 0, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that <math> r = \sqrt{\phi}</math> it generates a right triangle that is always similar to the Kepler triangle.

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have <math> b^2 = h^2 + d^2 </math> and <math> a^2 = h^2 + (c - d)^2 </math> according to the figure at the right. Subtracting these yields <math> a^2 - b^2 = c^2 - 2cd </math>. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that <math> h^2 = b^2 - d^2 </math>. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., ''x'' = 0 or ''y'' = 0, or
''x'' = ''α y'' for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote ''x'' = (1, 0) and ''y'' = (0, 1). Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, <math> d(x_n, x_m) \leq d(x_n, x) + d(x_m, x) < \epsilon /2 + \epsilon /2 = \epsilon </math>, so that the sequence {{math|{''xn''}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T22:31:11Z

Khanh: /* Converse */

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in {{closed-closed|0, ''π''}}) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides <math>\overline{AB} = \overline{AC}</math> divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides <math>\overline{AB} = \overline{AC}</math>. From the triangle postulate, the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But <math>\overline{AB} = \overline{AC}</math>. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows <math> \overline{AC} > \overline{DC}</math>, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the triangle postulate), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that ''d'' = ''a''/3, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation ''r''2 + ''r'' − 1 = 0, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation
''r''2 − ''r'' − 1 = 0, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that <math> r = \sqrt{\phi}</math> it generates a right triangle that is always similar to the Kepler triangle.

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have <math> b^2 = h^2 + d^2 </math> and <math> a^2 = h^2 + (c - d)^2 </math> according to the figure at the right. Subtracting these yields <math> a^2 - b^2 = c^2 - 2cd </math>. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that <math> h^2 = b^2 - d^2 </math>. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., ''x'' = 0 or ''y'' = 0, or
''x'' = ''α y'' for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote ''x'' = (1, 0) and ''y'' = (0, 1). Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, <math> d(x_n, x_m) \leq d(x_n, x) + d(x_m, x) < \epsilon /2 + \epsilon /2 = \epsilon </math>, so that the sequence {{math|{''xn''}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T22:28:49Z

Khanh: /* Converse */

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in {{closed-closed|0, ''π''}}) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides <math>\overline{AB} = \overline{AC}</math> divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides <math>\overline{AB} = \overline{AC}</math>. From the triangle postulate, the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But <math>\overline{AB} = \overline{AC}</math>. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows <math> \overline{AC} > \overline{DC}</math>, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the triangle postulate), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that ''d'' = ''a''/3, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation ''r''2 + ''r'' − 1 = 0, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation
''r''2 − ''r'' − 1 = 0, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that <math> r = \sqrt{\phi}</math> it generates a right triangle that is always similar to the Kepler triangle.

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have <math> b^2 = h^2 + d^2 </math> and <math> a^2 = h^2 + (c - d)^2 </math> according to the figure at the right. Subtracting these yields <math> a^2 - b^2 = c^2 − 2cd </math>. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that <math> h^2 = b^2 - d^2 </math>. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., ''x'' = 0 or ''y'' = 0, or
''x'' = ''α y'' for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote ''x'' = (1, 0) and ''y'' = (0, 1). Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, <math> d(x_n, x_m) \leq d(x_n, x) + d(x_m, x) < \epsilon /2 + \epsilon /2 = \epsilon </math>, so that the sequence {{math|{''xn''}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T22:21:34Z

Khanh: /* Examples of use */

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in {{closed-closed|0, ''π''}}) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides <math>\overline{AB} = \overline{AC}</math> divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides <math>\overline{AB} = \overline{AC}</math>. From the triangle postulate, the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But <math>\overline{AB} = \overline{AC}</math>. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows <math> \overline{AC} > \overline{DC}</math>, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the triangle postulate), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that ''d'' = ''a''/3, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation ''r''2 + ''r'' − 1 = 0, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation
''r''2 − ''r'' − 1 = 0, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that <math> r = \sqrt{\phi}</math> it generates a right triangle that is always similar to the Kepler triangle.

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have {{math|''b''{{sup|2}} {{=}} ''h''{{sup|2}} + ''d''{{sup|2}}}} and {{math|''a''{{sup|2}} {{=}} ''h''{{sup|2}} + (''c'' − ''d''){{sup|2}}}} according to the figure at the right. Subtracting these yields {{math|''a''{{sup|2}} − ''b''{{sup|2}} {{=}} ''c''{{sup|2}} − 2''cd''}}. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that {{math|''h''{{sup|2}} {{=}} ''b''{{sup|2}} − ''d''{{sup|2}}}}. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., ''x'' = 0 or ''y'' = 0, or
''x'' = ''α y'' for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote ''x'' = (1, 0) and ''y'' = (0, 1). Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, <math> d(x_n, x_m) \leq d(x_n, x) + d(x_m, x) < \epsilon /2 + \epsilon /2 = \epsilon </math>, so that the sequence {{math|{''xn''}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T22:09:51Z

Khanh: /* Right triangle */

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in {{closed-closed|0, ''π''}}) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides <math>\overline{AB} = \overline{AC}</math> divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides <math>\overline{AB} = \overline{AC}</math>. From the triangle postulate, the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But <math>\overline{AB} = \overline{AC}</math>. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows <math> \overline{AC} > \overline{DC}</math>, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the triangle postulate), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that {{math|''d'' {{=}} ''a''/3}}, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation {{math|''r''2 + ''r'' − 1 {{=}} 0}}, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation {{math|''r''2 − ''r'' − 1 {{=}} 0}}, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that {{math|''r'' {{=}} {{sqrt|''φ''}}}} it generates a right triangle that is always similar to the [[Kepler triangle]].

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have {{math|''b''{{sup|2}} {{=}} ''h''{{sup|2}} + ''d''{{sup|2}}}} and {{math|''a''{{sup|2}} {{=}} ''h''{{sup|2}} + (''c'' − ''d''){{sup|2}}}} according to the figure at the right. Subtracting these yields {{math|''a''{{sup|2}} − ''b''{{sup|2}} {{=}} ''c''{{sup|2}} − 2''cd''}}. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that {{math|''h''{{sup|2}} {{=}} ''b''{{sup|2}} − ''d''{{sup|2}}}}. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., ''x'' = 0 or ''y'' = 0, or
''x'' = ''α y'' for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote ''x'' = (1, 0) and ''y'' = (0, 1). Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, <math> d(x_n, x_m) \leq d(x_n, x) + d(x_m, x) < \epsilon /2 + \epsilon /2 = \epsilon </math>, so that the sequence {{math|{''xn''}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T22:07:51Z

Khanh: /* Right triangle */

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in {{closed-closed|0, ''π''}}) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides {{math|{{overline|AB}} {{=}} {{overline|AC}}}} divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides <math>\overline{AB} = \overline{AC}</math>. From the triangle postulate, the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But <math>\overline{AB} = \overline{AC}</math>. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows <math> \overline{AC} > \overline{DC}</math>, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the [[triangle postulate]]), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that {{math|''d'' {{=}} ''a''/3}}, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation {{math|''r''2 + ''r'' − 1 {{=}} 0}}, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation {{math|''r''2 − ''r'' − 1 {{=}} 0}}, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that {{math|''r'' {{=}} {{sqrt|''φ''}}}} it generates a right triangle that is always similar to the [[Kepler triangle]].

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have {{math|''b''{{sup|2}} {{=}} ''h''{{sup|2}} + ''d''{{sup|2}}}} and {{math|''a''{{sup|2}} {{=}} ''h''{{sup|2}} + (''c'' − ''d''){{sup|2}}}} according to the figure at the right. Subtracting these yields {{math|''a''{{sup|2}} − ''b''{{sup|2}} {{=}} ''c''{{sup|2}} − 2''cd''}}. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that {{math|''h''{{sup|2}} {{=}} ''b''{{sup|2}} − ''d''{{sup|2}}}}. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., ''x'' = 0 or ''y'' = 0, or
''x'' = ''α y'' for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote ''x'' = (1, 0) and ''y'' = (0, 1). Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, <math> d(x_n, x_m) \leq d(x_n, x) + d(x_m, x) < \epsilon /2 + \epsilon /2 = \epsilon </math>, so that the sequence {{math|{''xn''}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T22:04:51Z

Khanh: /* Normed vector space */

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in {{closed-closed|0, ''π''}}) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides {{math|{{overline|AB}} {{=}} {{overline|AC}}}} divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides {{math|{{overline|AB}} {{=}} {{overline|AC}}}}. From the [[triangle postulate]], the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But {{math|{{overline|AB}} {{=}} {{overline|AC}}}}. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows {{math|{{overline|AC}} > {{overline|DC}}}}, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the [[triangle postulate]]), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that {{math|''d'' {{=}} ''a''/3}}, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation {{math|''r''2 + ''r'' − 1 {{=}} 0}}, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation {{math|''r''2 − ''r'' − 1 {{=}} 0}}, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that {{math|''r'' {{=}} {{sqrt|''φ''}}}} it generates a right triangle that is always similar to the [[Kepler triangle]].

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have {{math|''b''{{sup|2}} {{=}} ''h''{{sup|2}} + ''d''{{sup|2}}}} and {{math|''a''{{sup|2}} {{=}} ''h''{{sup|2}} + (''c'' − ''d''){{sup|2}}}} according to the figure at the right. Subtracting these yields {{math|''a''{{sup|2}} − ''b''{{sup|2}} {{=}} ''c''{{sup|2}} − 2''cd''}}. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that {{math|''h''{{sup|2}} {{=}} ''b''{{sup|2}} − ''d''{{sup|2}}}}. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., ''x'' = 0 or ''y'' = 0, or
''x'' = ''α y'' for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote ''x'' = (1, 0) and ''y'' = (0, 1). Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, <math> d(x_n, x_m) \leq d(x_n, x) + d(x_m, x) < \epsilon /2 + \epsilon /2 = \epsilon </math>, so that the sequence {{math|{''xn''}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T21:59:11Z

Khanh: /* Metric space */

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in {{closed-closed|0, ''π''}}) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides {{math|{{overline|AB}} {{=}} {{overline|AC}}}} divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides {{math|{{overline|AB}} {{=}} {{overline|AC}}}}. From the [[triangle postulate]], the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But {{math|{{overline|AB}} {{=}} {{overline|AC}}}}. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows {{math|{{overline|AC}} > {{overline|DC}}}}, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the [[triangle postulate]]), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that {{math|''d'' {{=}} ''a''/3}}, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation {{math|''r''2 + ''r'' − 1 {{=}} 0}}, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation {{math|''r''2 − ''r'' − 1 {{=}} 0}}, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that {{math|''r'' {{=}} {{sqrt|''φ''}}}} it generates a right triangle that is always similar to the [[Kepler triangle]].

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have {{math|''b''{{sup|2}} {{=}} ''h''{{sup|2}} + ''d''{{sup|2}}}} and {{math|''a''{{sup|2}} {{=}} ''h''{{sup|2}} + (''c'' − ''d''){{sup|2}}}} according to the figure at the right. Subtracting these yields {{math|''a''{{sup|2}} − ''b''{{sup|2}} {{=}} ''c''{{sup|2}} − 2''cd''}}. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that {{math|''h''{{sup|2}} {{=}} ''b''{{sup|2}} − ''d''{{sup|2}}}}. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., {{math|''x'' {{=}} 0}} or {{math|''y'' {{=}} 0}}, or
{{math|''x'' {{=}} ''α y''}} for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote {{math|''x'' {{=}} (1, 0)}} and {{math|''y'' {{=}} (0, 1)}}. Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in [[mathematical analysis]] for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, <math> d(x_n, x_m) \leq d(x_n, x) + d(x_m, x) < \epsilon /2 + \epsilon /2 = \epsilon </math>, so that the sequence {{math|{''xn''}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T21:50:32Z

Khanh: /* Euclidean geometry */

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in {{closed-closed|0, ''π''}}) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as <math>\overline{BC}</math> and the other equal leg <math>\overline{BD}</math> along the extension of side <math>\overline{AB}</math>. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side <math>\overline{AD}</math> is longer than side <math>\overline{AC}</math>. But <math> AD = AB + BD = AB + BC, so the sum of the lengths of sides <math>\overline{AB}</math> and <math>\overline{BC}</math> is larger than the length of <math>\overline{AC}</math>. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the golden ratio, as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides {{math|{{overline|AB}} {{=}} {{overline|AC}}}} divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides {{math|{{overline|AB}} {{=}} {{overline|AC}}}}. From the [[triangle postulate]], the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But {{math|{{overline|AB}} {{=}} {{overline|AC}}}}. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows {{math|{{overline|AC}} > {{overline|DC}}}}, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the [[triangle postulate]]), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that {{math|''d'' {{=}} ''a''/3}}, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation {{math|''r''2 + ''r'' − 1 {{=}} 0}}, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation {{math|''r''2 − ''r'' − 1 {{=}} 0}}, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that {{math|''r'' {{=}} {{sqrt|''φ''}}}} it generates a right triangle that is always similar to the [[Kepler triangle]].

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have {{math|''b''{{sup|2}} {{=}} ''h''{{sup|2}} + ''d''{{sup|2}}}} and {{math|''a''{{sup|2}} {{=}} ''h''{{sup|2}} + (''c'' − ''d''){{sup|2}}}} according to the figure at the right. Subtracting these yields {{math|''a''{{sup|2}} − ''b''{{sup|2}} {{=}} ''c''{{sup|2}} − 2''cd''}}. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that {{math|''h''{{sup|2}} {{=}} ''b''{{sup|2}} − ''d''{{sup|2}}}}. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., {{math|''x'' {{=}} 0}} or {{math|''y'' {{=}} 0}}, or
{{math|''x'' {{=}} ''α y''}} for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote {{math|''x'' {{=}} (1, 0)}} and {{math|''y'' {{=}} (0, 1)}}. Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in [[mathematical analysis]] for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, {{math|''d''(''xn'', ''xm'') ≤ ''d''(''xn'', ''x'') + ''d''(''xm'', ''x'') < ''ε''/2 + ''ε''/2 {{=}} ''ε''}}, so that the sequence {{math|{{mset|''xn''}}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Triangle Inequality

2022-02-13T21:40:26Z

Khanh:

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{math|''x''}}, {{math|''y''}}, {{math|''z''}}. The top example shows a case where {{math|''z''}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides, and the bottom example shows a case where the side {{math|''z''}} is only slightly less than {{math|''x'' + ''y''}}.]]

In mathematics, the '''triangle inequality''' states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the lengths of the sides of the triangle, with no side being greater than {{math|''z''}}, then the triangle inequality states that
:<math>z \leq x + y ,</math>
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
:<math>\|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| ,</math>
where the length {{math|''z''}} of the third side has been replaced by the vector sum {{math|'''x''' + '''y'''}}. When {{math|'''x'''}} and {{math|'''y'''}} are real numbers, they can be viewed as vectors in {{math|'''R'''1}}, and the triangle inequality expresses a relationship between absolute values.

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either {{math|'''R'''2}} or {{math|'''R'''3}}. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a {{math|180°}} angle and two {{math|0°}} angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in {{closed-closed|0, ''π''}}) with those endpoints.

The triangle inequality is a ''defining property'' of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces ({{math|''p'' ≥ 1}}), and inner product spaces.

==Euclidean geometry==

[[File:Euclid triangle inequality.svg|thumb|Euclid's construction for proof of the triangle inequality for plane geometry.]]

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle {{math|''ABC''}}, an isosceles triangle is constructed with one side taken as {{math|{{overline|''BC''}}}} and the other equal leg {{math|{{overline|''BD''}}}} along the extension of side {{math|{{overline|''AB''}}}}. It then is argued that angle {{math|''β''}} has larger measure than angle {{math|''α''}}, so side {{math|{{overline|''AD''}}}} is longer than side {{math|{{overline|''AC''}}}}. But {{math|''AD'' {{=}} ''AB'' + ''BD'' {{=}} ''AB'' + ''BC''}}, so the sum of the lengths of sides {{math|{{overline|''AB''}}}} and {{math|{{overline|''BC''}}}} is larger than the length of {{math|{{overline|''AC''}}}}. This proof appears in Euclid's Elements, Book 1, Proposition 20.

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a''}}, {{math|''b''}}, {{math|''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero. Heron's formula for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and <math>\phi</math> is the [[golden ratio]], as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math>

===Right triangle===

[[File:Isosceles triangle made of right triangles.svg|thumb|Isosceles triangle with equal sides {{math|{{overline|AB}} {{=}} {{overline|AC}}}} divided into two right triangles by an altitude drawn from one of the two base angles.]]

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle {{math|ADC}}. An isosceles triangle {{math|ABC}} is constructed with equal sides {{math|{{overline|AB}} {{=}} {{overline|AC}}}}. From the [[triangle postulate]], the angles in the right triangle {{math|ADC}} satisfy:
:<math> \alpha + \gamma = \pi /2 \ . </math>
Likewise, in the isosceles triangle {{math|ABC}}, the angles satisfy:
:<math>2\beta + \gamma = \pi \ . </math>
Therefore,
:<math> \alpha = \pi/2 - \gamma ,\ \mathrm{while} \ \beta= \pi/2 - \gamma /2 \ ,</math>
and so, in particular,
:<math>\alpha < \beta \ . </math>
That means side {{math|AD}} opposite angle {{math|''α''}} is shorter than side {{math|AB}} opposite the larger angle {{math|''β''}}. But {{math|{{overline|AB}} {{=}} {{overline|AC}}}}. Hence:
:<math>\overline{\mathrm{AC}} > \overline{\mathrm{AD}} \ . </math>
A similar construction shows {{math|{{overline|AC}} > {{overline|DC}}}}, establishing the theorem.

An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point {{math|B}}:
(i) as depicted (which is to be proven), or (ii) {{math|B}} coincident with {{math|D}} (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle {{math|''γ''}}, which would violate the [[triangle postulate]]), or lastly, (iii) {{math|B}} interior to the right triangle between points {{math|A}} and {{math|D}} (in which case angle {{math|ABC}} is an exterior angle of a right triangle {{math|BDC}} and therefore larger than {{math|''π''/2}}, meaning the other base angle of the isosceles triangle also is greater than {{math|''π''/2}} and their sum exceeds {{math|''π''}} in violation of the triangle postulate).

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

===Examples of use===
Consider a triangle whose sides are in an arithmetic progression and let the sides be {{math|''a''}}, {{math|''a'' + ''d''}}, {{math|''a'' + 2''d''}}. Then the triangle inequality requires that

:<math> 0<a<2a+3d </math>
:<math> 0<a+d<2a+2d </math>
:<math> 0<a+2d<2a+d. </math>

To satisfy all these inequalities requires

:<math> a>0 \text{ and } -\frac{a}{3}<d<a. </math>

When {{math|''d''}} is chosen such that {{math|''d'' {{=}} ''a''/3}}, it generates a right triangle that is always similar to the Pythagorean triple with sides {{math|3}}, {{math|4}}, {{math|5}}.

Now consider a triangle whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}. Then the triangle inequality requires that

:<math> 0<a<ar+ar^2 </math>
:<math> 0<ar<a+ar^2 </math>
:<math> 0<ar^2<a+ar. </math>

The first inequality requires {{math|''a'' > 0}}; consequently it can be divided through and eliminated. With {{math|''a'' > 0}}, the middle inequality only requires {{math|''r'' > 0}}. This now leaves the first and third inequalities needing to satisfy

:<math>
\begin{align}
r^2+r-1 & {} >0 \\
r^2-r-1 & {} <0.
\end{align}
</math>

The first of these quadratic inequalities requires {{math|''r''}} to range in the region beyond the value of the positive root of the quadratic equation {{math|''r''2 + ''r'' − 1 {{=}} 0}}, i.e. {{math|''r'' > ''φ'' − 1}} where {{math|''φ''}} is the golden ratio. The second quadratic inequality requires {{math|''r''}} to range between 0 and the positive root of the quadratic equation {{math|''r''2 − ''r'' − 1 {{=}} 0}}, i.e. {{math|0 < ''r'' < ''φ''}}. The combined requirements result in {{math|''r''}} being confined to the range
:<math>\varphi - 1 < r <\varphi\, \text{ and } a >0.</math>

When {{math|''r''}} the common ratio is chosen such that {{math|''r'' {{=}} {{sqrt|''φ''}}}} it generates a right triangle that is always similar to the [[Kepler triangle]].

===Generalization to any polygon===
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a geometric progression and let the sides be {{math|''a''}}, {{math|''ar''}}, {{math|''ar''2}}, {{math|''ar''3}}. Then the generalized polygon inequality requires that

:<math> 0<a<ar+ar^2+ar^3 </math>
:<math> 0<ar<a+ar^2+ar^3 </math>
:<math> 0<ar^2<a+ar+ar^3 </math>
:<math> 0<ar^3<a+ar+ar^2. </math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math>
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, {{math|''r''}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{math|''t''}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{math|''h''}} cutting base {{math|''c''}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the Pythagorean theorem we have {{math|''b''{{sup|2}} {{=}} ''h''{{sup|2}} + ''d''{{sup|2}}}} and {{math|''a''{{sup|2}} {{=}} ''h''{{sup|2}} + (''c'' − ''d''){{sup|2}}}} according to the figure at the right. Subtracting these yields {{math|''a''{{sup|2}} − ''b''{{sup|2}} {{=}} ''c''{{sup|2}} − 2''cd''}}. This equation allows us to express {{math|''d''}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that {{math|''h''{{sup|2}} {{=}} ''b''{{sup|2}} − ''d''{{sup|2}}}}. By replacing {{math|''d''}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number ''h'' to satisfy this, <math>h^2</math> must be non-negative:
:<math>b^2-\left (\frac{-a^2+b^2+c^2}{2c}\right) ^2 \ge 0,</math>
:<math>\left( b- \frac{-a^2+b^2+c^2}{2c}\right) \left( b+ \frac{-a^2+b^2+c^2}{2c}\right) \ge 0,</math>
:<math>\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0,</math>
:<math>(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0,</math>
:<math>(a+b-c)(a+c-b)(b+c-a) \ge 0,</math>
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.

===Generalization to higher dimensions===

The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-facet of an {{math|''n''}}-simplex is less than or equal to the sum of the hypervolumes of the other {{math|''n''}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 7}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 4}}.
However, points with such distances cannot exist: the area of the 7-7-7 equilateral triangle {{math|''ABC''}} would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a normed vector space {{math|''V''}}, one of the defining properties of the norm is the triangle inequality:

:<math> \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V</math>

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then <math>\|x+y\|=\|x\|+\|y\|</math> if and
only if the triangle formed by {{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is degenerate, that is,
{{math|''x''}} and {{math|''y''}} are on the same ray, i.e., {{math|''x'' {{=}} 0}} or {{math|''y'' {{=}} 0}}, or
{{math|''x'' {{=}} ''α y''}} for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as
the {{math|''ℓp''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is
not true. For instance, consider the plane with the {{math|''ℓ''1}} norm (the Manhattan distance) and
denote {{math|''x'' {{=}} (1, 0)}} and {{math|''y'' {{=}} (0, 1)}}. Then the triangle formed by
{{math|''x''}}, {{math|''y''}}, and {{math|''x'' + ''y''}}, is non-degenerate but

:<math>\|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.</math>

===Example norms===
*''Absolute value as norm for the real line.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers {{math|''x''}} and {{math|''y''}}: <math display="block">|x + y| \leq |x|+|y|,</math> which it does.

Proof:

:<math>-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert</math>
:<math>-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert</math>
After adding,
:<math>-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert</math>
Use the fact that <math>\left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a</math>
(with ''b'' replaced by ''x''+''y'' and ''a'' by <math>\left\vert x \right\vert + \left\vert y \right\vert</math>), we have

:<math>|x + y| \leq |x|+|y|</math>

The triangle inequality is useful in [[mathematical analysis]] for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{math|''x''}} and {{math|''y''}}:

:<math>|x-y| \geq \biggl||x|-|y|\biggr|.</math>

*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors <math>x</math> and <math>y</math>, and denoting the inner product as <math>\langle x , y\rangle </math>:
:{|
|<math>\|x + y\|^2</math> || <math>= \langle x + y, x + y \rangle</math>
|-
| || <math>= \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math>
|-
| || <math>\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>= \left(\|x\| + \|y\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{math|''x''}} and {{math|''y''}}
are linearly dependent. The inequality
<math>\langle x, y \rangle + \langle y, x \rangle \le 2\left|\left\langle x, y \right\rangle\right| </math>
turns into an equality for linearly dependent <math>x</math> and <math>y</math>
if and only if one of the vectors {{math|''x''}} or {{math|''y''}} is a ''nonnegative'' scalar of the other.

:Taking the square root of the final result gives the triangle inequality.
*{{math|''p''}}-norm: a commonly used norm is the ''p''-norm: <math display="block">\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right) ^{1/p} \ , </math> where the {{math|''xi''}} are the components of vector {{math|''x''}}. For {{math|1=''p'' = 2}} the {{math|''p''}}-norm becomes the ''Euclidean norm'': <math display="block">\|x\|_2 = \left( \sum_{i=1}^n |x_i|^2 \right) ^{1/2} = \left( \sum_{i=1}^n x_{i}^2 \right) ^{1/2} \ , </math> which is Pythagoras' theorem in {{math|''n''}}-dimensions, a very special case corresponding to an inner product norm. Except for the case {{math|1=''p'' = 2}}, the {{math|''p''}}-norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of {{math|''p''}} is called Minkowski's inequality. It takes the form:<math display="block">\|x+y\|_p \le \|x\|_p + \|y\|_p \ .</math>

==Metric space==
In a metric space {{math|''M''}} with metric {{math|''d''}}, the triangle inequality is a requirement upon distance:
:<math>d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , </math>

for all {{math|''x''}}, {{math|''y''}}, {{math|''z''}} in {{math|''M''}}. That is, the distance from {{math|''x''}} to {{math|''z''}} is at most as large as the sum of the distance from {{math|''x''}} to {{math|''y''}} and the distance from {{math|''y''}} to {{math|''z''}}.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any {{math|''xn''}} and {{math|''xm''}} such that {{math|''d''(''xn'', ''x'') < ''ε''/2}} and {{math|''d''(''xm'', ''x'') < ''ε''/2}}, where {{math|''ε'' > 0}} is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, {{math|''d''(''xn'', ''xm'') ≤ ''d''(''xn'', ''x'') + ''d''(''xm'', ''x'') < ''ε''/2 + ''ε''/2 {{=}} ''ε''}}, so that the sequence {{math|{{mset|''xn''}}}} is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via {{math|''d''(''x'', ''y'') ≔ ‖''x'' − ''y''‖}}, with {{math|''x'' − ''y''}} being the vector pointing from point {{math|''y''}} to {{math|''x''}}.

==Reverse triangle inequality==
The '''reverse triangle inequality''' is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:

:''Any side of a triangle is greater than or equal to the difference between the other two sides''.

In the case of a normed vector space, the statement is:
: <math>\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,</math>
or for metric spaces, {{math|{{!}}''d''(''y'', ''x'') − ''d''(''x'', ''z''){{!}} ≤ ''d''(''y'', ''z'')}}.
This implies that the norm <math>\|\cdot\|</math> as well as the distance function <math>d(x,\cdot)</math> are Lipschitz continuous with Lipschitz constant {{math|1}}, and therefore are in particular uniformly continuous.

The proof for the reverse triangle uses the regular triangle inequality, and <math> \|y-x\| = \|{-}1(x-y)\| = |{-}1|\cdot\|x-y\| = \|x-y\| </math>:
: <math> \|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\| \Rightarrow \|x\| - \|y\| \leq \|x-y\|, </math>
: <math> \|y\| = \|(y-x) + x\| \leq \|y-x\| + \|x\| \Rightarrow \|x\| - \|y\| \geq -\|x-y\|, </math>

Combining these two statements gives:
: <math> -\|x-y\| \leq \|x\|-\|y\| \leq \|x-y\| \Rightarrow \bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|.</math>

==Triangle inequality for cosine similarity==
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that

<math display="block">\operatorname{sim}(x,z) \geq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) - \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}</math>

and

<math display="block">\operatorname{sim}(x,z) \leq \operatorname{sim}(x,y) \cdot \operatorname{sim}(y,z) + \sqrt{\left(1-\operatorname{sim}(x,y)^2\right)\cdot\left(1-\operatorname{sim}(y,z)^2\right)}\,.</math>

With these formulas, one needs to compute a square root for each triple of vectors {{math|{''x'', ''y'', ''z''}}} that is examined rather than {{math|arccos(sim(''x'',''y''))}} for each pair of vectors {{math|{''x'', ''y''}}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.

==Reversal in Minkowski space==

The Minkowski space metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|x\|^2 = \eta_{\mu \nu} x^\mu x^\nu</math> can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|x+y\| \geq \|x\| + \|y\|. </math>

A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Triangle_inequality Triangle inequality, Wikipedia] under a CC BY-SA license

Area of Polygons - Formulas

2022-02-07T00:00:42Z

Khanh:

== Area of Parallelogram ==

[[File:ParallelogramArea.svg|thumb|180px|alt=A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle|A parallelogram can be rearranged into a rectangle with the same area.]]
[[File:Parallelogram area animated.gif|thumb|180px|Animation for the area formula <math>K = bh</math>.]]
All of the area formulas for general convex quadrilaterals apply to parallelograms. Further formulas are specific to parallelograms:

A parallelogram with base ''b'' and height ''h'' can be divided into a trapezoid and a right triangle, and rearranged into a rectangle, as shown in the figure to the left. This means that the area of a parallelogram is the same as that of a rectangle with the same base and height:

:<math>K = bh.</math>

[[File:Parallelogram area.svg|thumb|The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram]]
The base × height area formula can also be derived using the figure to the right. The area ''K'' of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is

:<math>K_\text{rect} = (B+A) \times H\,</math>

and the area of a single orange triangle is

:<math>K_\text{tri} = \frac{A}{2} \times H. \,</math>

Therefore, the area of the parallelogram is

:<math>K = K_\text{rect} - 2 \times K_\text{tri} = ( (B+A) \times H) - ( A \times H) = B \times H.</math>

Another area formula, for two sides ''B'' and ''C'' and angle θ, is

:<math>K = B \cdot C \cdot \sin \theta.\,</math>

The area of a parallelogram with sides ''B'' and ''C'' (''B'' ≠ ''C'') and angle <math>\gamma</math> at the intersection of the diagonals is given by

:<math>K = \frac{|\tan \gamma|}{2} \cdot \left| B^2 - C^2 \right|.</math>

When the parallelogram is specified from the lengths ''B'' and ''C'' of two adjacent sides together with the length ''D''1 of either diagonal, then the area can be found from Heron's formula. Specifically it is

:<math>K=2\sqrt{S(S-B)(S-C)(S-D_1)}</math>

where <math>S=(B+C+D_1)/2</math> and the leading factor 2 comes from the fact that the chosen diagonal divides the parallelogram into ''two'' congruent triangles.

== Area of Triangle ==
Calculating the area ''T'' of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is:

<math> T=\frac {1}{2}bh </math>,

where b is the length of the base of the triangle, and ''h'' is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the base onto the line containing the base. In 499 CE Aryabhata, used this illustrated method in the Aryabhatiya (section 2.6).

Although simple, this formula is only useful if the height can be readily found, which is not always the case. For example, the surveyor of a triangular field might find it relatively easy to measure the length of each side, but relatively difficult to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle.

=== Using trigonometry ===

Applying trigonometry to find the altitude h.
The height of a triangle can be found through the application of trigonometry.

''Knowing SAS:'' Using the labels in the image on the right, the altitude is <math> h = a \sin {\gamma }</math>. Substituting this in the formula <math> T=\frac {1}{2}bh </math> derived above, the area of the triangle can be expressed as:

<math> T=\frac {1}{2} ab \sin {\gamma} =\frac {1}{2} bc \sin {\alpha} = \frac {1}{2} ca \sin {\beta} </math>

(where α is the interior angle at ''A'', β is the interior angle at ''B'', <math> \gamma </math> is the interior angle at ''C'' and ''c'' is the line '''AB''').

Furthermore, since sin α = sin (π − α) = sin (β + <math> \gamma </math>), and similarly for the other two angles:

<math> T=\frac {1}{2} ab \sin(\alpha +\beta)= \frac {1}{2} bc \sin(\beta +\gamma)= \frac {1}{2} ca \sin(\gamma +\alpha)</math>.

''Knowing AAS:''

<math> T= \frac {b^{2}(\sin \alpha )(\sin(\alpha +\beta ))}{2\sin \beta} </math>,
and analogously if the known side is ''a'' or ''c''.

''Knowing ASA'':

<math> T= \frac {a^2}{2(\cot \beta +\cot \gamma )}= \frac {a^2 (\sin \beta ) (\sin \gamma )}{2\sin(\beta + \gamma)} </math>,
and analogously if the known side is ''b'' or ''c''.

=== Using Heron's formula ===
The shape of the triangle is determined by the lengths of the sides. Therefore, the area can also be derived from the lengths of the sides. By Heron's formula:

<math> T= \sqrt {s(s-a)(s-b)(s-c)} </math>
where <math> s= \tfrac {a+b+c}{2} </math> is the semiperimeter, or half of the triangle's perimeter.

Three other equivalent ways of writing Heron's formula are

<math> T= \frac {1}{4} \sqrt {(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})} </math>

<math> T= \frac {1}{4} \sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})} </math>

<math> T= \frac {1}{4} \sqrt {(a+b-c)(a-b+c)(-a+b+c)(a+b+c)} </math>.

== Area of Trapezoid ==

The area ''K'' of a trapezoid is given by
:<math>K = \frac{a + b}{2} \cdot h = mh</math>

where ''a'' and ''b'' are the lengths of the parallel sides, ''h'' is the height (the perpendicular distance between these sides), and ''m'' is the arithmetic mean of the lengths of the two parallel sides. In 499 AD Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the ''Aryabhatiya'' (section 2.8). This yields as a special case the well-known formula for the area of a triangle, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.

The 7th-century Indian mathematician Bhāskara I derived the following formula for the area of a trapezoid with consecutive sides ''a'', ''c'', ''b'', ''d'':
:<math>K=\frac{1}{2}(a+b)\sqrt{c^2-\frac{1}{4}\left((b-a)+\frac{c^2-d^2}{b-a}\right)^2}</math>

where ''a'' and ''b'' are parallel and ''b'' > ''a''. This formula can be factored into a more symmetric version
:<math>K = \frac{a+b}{4|b-a|}\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}.</math>

When one of the parallel sides has shrunk to a point (say ''a'' = 0), this formula reduces to Heron's formula for the area of a triangle.

Another equivalent formula for the area, which more closely resembles Heron's formula, is
:<math>K = \frac{a+b}{|b-a|}\sqrt{(s-b)(s-a)(s-b-c)(s-b-d)},</math>

where <math>s = \tfrac{1}{2}(a + b + c + d)</math> is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula, but it differs from it, in that a trapezoid might not be cyclic (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral).

From Bretschneider's formula, it follows that
:<math>K= \sqrt{\frac{(ab^2-a^2 b-ad^2+bc^2)(ab^2-a^2 b-ac^2+bd^2)}{(2(b-a))^2} - \left(\frac{b^2+d^2-a^2-c^2}{4}\right)^2}.</math>

The line that joins the midpoints of the parallel sides, bisects the area.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Parallelogram Parallelogram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Triangle Triangle, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Trapezoid Trapezoid, Wikipedia] under a CC BY-SA license

Measurement (AREA)

2022-02-06T17:58:10Z

Khanh:

[[File:Area.svg|right|thumb|alt=Three shapes on a square grid|The combined area of these three shapes is approximately 15.57 squares.]]

'''Area''' is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.
[[File:Squaring the circle.svg|right|thumb|This square and this disk both have the same area (see: squaring the circle).]]

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

==Formal definition==
An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties:
* For all ''S'' in ''M'', ''a''(''S'') ≥ 0.
* If ''S'' and ''T'' are in ''M'' then so are ''S'' ∪ ''T'' and ''S'' ∩ ''T'', and also ''a''(''S''∪''T'') = ''a''(''S'') + ''a''(''T'') − ''a''(''S''∩''T'').
* If ''S'' and ''T'' are in ''M'' with ''S'' ⊆ ''T'' then ''T'' − ''S'' is in ''M'' and ''a''(''T''−''S'') = ''a''(''T'') − ''a''(''S'').
* If a set ''S'' is in ''M'' and ''S'' is congruent to ''T'' then ''T'' is also in ''M'' and ''a''(''S'') = ''a''(''T'').
* Every rectangle ''R'' is in ''M''. If the rectangle has length ''h'' and breadth ''k'' then ''a''(''R'') = ''hk''.
* Let ''Q'' be a set enclosed between two step regions ''S'' and ''T''. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. ''S'' ⊆ ''Q'' ⊆ ''T''. If there is a unique number ''c'' such that ''a''(''S'') ≤ c ≤ ''a''(''T'') for all such step regions ''S'' and ''T'', then ''a''(''Q'') = ''c''.

It can be proved that such an area function actually exists.

==Units==
[[File:SquareMeterQuadrat.JPG|thumb|right|alt=A square made of PVC pipe on grass|A square metre quadrat made of PVC pipe.]]
Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m2), square centimetres (cm2), square millimetres (mm2), square kilometres (km2), square feet (ft2), square yards (yd2), square miles (mi2), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.

The SI unit of area is the square metre, which is considered an SI derived unit.

===Conversions===
[[File:Area conversion - square mm in a square cm.png|thumb|right|alt=A diagram showing the conversion factor between different areas|Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.]]
Calculation of the area of a square whose length and width are 1 metre would be:

1 metre × 1 metre = 1 m2

and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as:

3 metres × 2 metres = 6 m2. This is equivalent to 6 million square millimetres. Other useful conversions are:
* 1 square kilometre = 1,000,000 square metres
* 1 square metre = 10,000 square centimetres = 1,000,000 square millimetres
* 1 square centimetre = 100 square millimetres.

==== Non-metric units ====
In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units.
:1 foot = 12 inches,
the relationship between square feet and square inches is
:1 square foot = 144 square inches,
where 144 = 122 = 12 × 12. Similarly:
* 1 square yard = 9 square feet
* 1 square mile = 3,097,600 square yards = 27,878,400 square feet
In addition, conversion factors include:
* 1 square inch = 6.4516 square centimetres
* 1 square foot = 0.092 903 04 square metres
* 1 square yard = 0.836 127 36 square metres
* 1 square mile = 2.589 988 110 336 square kilometres

===Other units including historical===
There are several other common units for area. The are was the original unit of area in the metric system, with:
* 1 are = 100 square metres
Though the are has fallen out of use, the hectare is still commonly used to measure land:
* 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres
Other uncommon metric units of area include the tetrad, the hectad, and the myriad.

The acre is also commonly used to measure land areas, where
* 1 acre = 4,840 square yards = 43,560 square feet.
An acre is approximately 40% of a hectare.

On the atomic scale, area is measured in units of barns, such that:
* 1 barn = 10−28 square meters.
The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics.

In India,
* 20 dhurki = 1 dhur
* 20 dhur = 1 khatha
* 20 khata = 1 bigha
* 32 khata = 1 acre

== Resources ==
* [https://mathbitsnotebook.com/Geometry/CoordinateGeometry/CGArea.html Area and Perimeter on a Grid, MathBitsNotebook]

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Area Area, Wikipedia] under a CC BY-SA license

Measurement (AREA)

2022-02-06T17:57:34Z

Khanh: Created page with "The combined area of these three shapes is approximately 15.57 squares. '''Area''' is the quantity that expres..."

[[File:Area.svg|right|thumb|alt=Three shapes on a square grid|The combined area of these three shapes is approximately 15.57 squares.]]

'''Area''' is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.
[[File:Squaring the circle.svg|right|thumb|This square and this disk both have the same area (see: squaring the circle).]]

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

==Formal definition==
An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties:
* For all ''S'' in ''M'', ''a''(''S'') ≥ 0.
* If ''S'' and ''T'' are in ''M'' then so are ''S'' ∪ ''T'' and ''S'' ∩ ''T'', and also ''a''(''S''∪''T'') = ''a''(''S'') + ''a''(''T'') − ''a''(''S''∩''T'').
* If ''S'' and ''T'' are in ''M'' with ''S'' ⊆ ''T'' then ''T'' − ''S'' is in ''M'' and ''a''(''T''−''S'') = ''a''(''T'') − ''a''(''S'').
* If a set ''S'' is in ''M'' and ''S'' is congruent to ''T'' then ''T'' is also in ''M'' and ''a''(''S'') = ''a''(''T'').
* Every rectangle ''R'' is in ''M''. If the rectangle has length ''h'' and breadth ''k'' then ''a''(''R'') = ''hk''.
* Let ''Q'' be a set enclosed between two step regions ''S'' and ''T''. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. ''S'' ⊆ ''Q'' ⊆ ''T''. If there is a unique number ''c'' such that ''a''(''S'') ≤ c ≤ ''a''(''T'') for all such step regions ''S'' and ''T'', then ''a''(''Q'') = ''c''.

It can be proved that such an area function actually exists.

==Units==
[[File:SquareMeterQuadrat.JPG|thumb|right|alt=A square made of PVC pipe on grass|A square metre quadrat made of PVC pipe.]]
Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m2), square centimetres (cm2), square millimetres (mm2), square kilometres (km2), square feet (ft2), square yards (yd2), square miles (mi2), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.

The SI unit of area is the square metre, which is considered an SI derived unit.

===Conversions===
[[File:Area conversion - square mm in a square cm.png|thumb|right|alt=A diagram showing the conversion factor between different areas|Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.]]
Calculation of the area of a square whose length and width are 1 metre would be:

1 metre × 1 metre = 1 m2

and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as:

3 metres × 2 metres = 6 m2. This is equivalent to 6 million square millimetres. Other useful conversions are:
* 1 square kilometre = 1,000,000 square metres
* 1 square metre = 10,000 square centimetres = 1,000,000 square millimetres
* 1 square centimetre = 100 square millimetres.

==== Non-metric units ====
In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units.
:1 foot = 12 inches,
the relationship between square feet and square inches is
:1 square foot = 144 square inches,
where 144 = 122 = 12 × 12. Similarly:
* 1 square yard = 9 square feet
* 1 square mile = 3,097,600 square yards = 27,878,400 square feet
In addition, conversion factors include:
* 1 square inch = 6.4516 square centimetres
* 1 square foot = 0.092 903 04 square metres
* 1 square yard = 0.836 127 36 square metres
* 1 square mile = 2.589 988 110 336 square kilometres

===Other units including historical===
There are several other common units for area. The are was the original unit of area in the metric system, with:
* 1 are = 100 square metres
Though the are has fallen out of use, the hectare is still commonly used to measure land:
* 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres
Other uncommon metric units of area include the tetrad, the hectad, and the myriad.

The acre is also commonly used to measure land areas, where
* 1 acre = 4,840 square yards = 43,560 square feet.
An acre is approximately 40% of a hectare.

On the atomic scale, area is measured in units of barns, such that:
* 1 barn = 10−28 square meters.
The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics.

In India,
* 20 dhurki = 1 dhur
* 20 dhur = 1 khatha
* 20 khata = 1 bigha
* 32 khata = 1 acre

== Resources ==
* [https://mathbitsnotebook.com/Geometry/CoordinateGeometry/CGArea.html Area and Perimeter on a Grid, MathBitsNotebook]
== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Area Area, Wikipedia] under a CC BY-SA license

Mathematical & Statistical Reasoning

2022-02-06T03:53:40Z

Khanh: /* Models and assumptions */

'''Statistical inference''' is the process of using data analysis to infer properties of an underlying distribution of probability. '''Inferential statistical analysis''' infers properties of a population, for example by '''testing hypotheses''' and deriving estimates. It is assumed that the observed data set is sampled from a larger population.

'''Inferential statistics''' can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. In machine learning, the term ''inference'' is sometimes used instead to mean "make a prediction, by evaluating an already trained model"; in this context inferring properties of the model is referred to as ''training'' or ''learning'' (rather than ''inference''), and using a model for prediction is referred to as ''inference'' (instead of ''prediction''); see also predictive inference.

== Introduction ==
Statistical inference makes propositions about a population, using data drawn from the population with some form of sampling. Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) selecting a statistical model of the process that generates the data and (second) deducing propositions from the model.

Konishi & Kitagawa state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling". Relatedly, Sir David Cox has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".

The conclusion of a statistical inference is a statistical proposition. Some common forms of statistical proposition are the following:
* a point estimate, i.e. a particular value that best approximates some parameter of interest;
* an interval estimate, e.g. a confidence interval (or set estimate), i.e. an interval constructed using a dataset drawn from a population so that, under repeated sampling of such datasets, such intervals would contain the true parameter value with the probability at the stated confidence level;
* a credible interval, i.e. a set of values containing, for example, 95% of posterior belief;
* rejection of a hypothesis;
* clustering or classification of data points into groups.

== Models and assumptions ==

Any statistical inference requires some assumptions. A '''statistical model''' is a set of assumptions concerning the generation of the observed data and similar data. Descriptions of statistical models usually emphasize the role of population quantities of interest, about which we wish to draw inference. Descriptive statistics are typically used as a preliminary step before more formal inferences are drawn.

=== Degree of models/assumptions ===
Statisticians distinguish between three levels of modeling assumptions;
* '''Fully parametric''': The probability distributions describing the data-generation process are assumed to be fully described by a family of probability distributions involving only a finite number of unknown parameters. For example, one may assume that the distribution of population values is truly Normal, with unknown mean and variance, and that datasets are generated by 'simple' random sampling. The family of generalized linear models is a widely used and flexible class of parametric models.
* '''Non-parametric''': The assumptions made about the process generating the data are much less than in parametric statistics and may be minimal. For example, every continuous probability distribution has a median, which may be estimated using the sample median or the Hodges–Lehmann–Sen estimator, which has good properties when the data arise from simple random sampling.
* '''Semi-parametric''': This term typically implies assumptions 'in between' fully and non-parametric approaches. For example, one may assume that a population distribution has a finite mean. Furthermore, one may assume that the mean response level in the population depends in a truly linear manner on some covariate (a parametric assumption) but not make any parametric assumption describing the variance around that mean (i.e. about the presence or possible form of any heteroscedasticity). More generally, semi-parametric models can often be separated into 'structural' and 'random variation' components. One component is treated parametrically and the other non-parametrically. The well-known Cox model is a set of semi-parametric assumptions.

=== Importance of valid models/assumptions ===

Whatever level of assumption is made, correctly calibrated inference, in general, requires these assumptions to be correct; i.e. that the data-generating mechanisms really have been correctly specified.

Incorrect assumptions of ' simple' random sampling can invalidate statistical inference. More complex semi- and fully parametric assumptions are also cause for concern. For example, incorrectly assuming the Cox model can in some cases lead to faulty conclusions. Incorrect assumptions of Normality in the population also invalidates some forms of regression-based inference. The use of '''any''' parametric model is viewed skeptically by most experts in sampling human populations: "most sampling statisticians, when they deal with confidence intervals at all, limit themselves to statements about [estimators] based on very large samples, where the central limit theorem ensures that these [estimators] will have distributions that are nearly normal." In particular, a normal distribution "would be a totally unrealistic and catastrophically unwise assumption to make if we were dealing with any kind of economic population." Here, the central limit theorem states that the distribution of the sample mean "for very large samples" is approximately normally distributed, if the distribution is not heavy-tailed.

====Approximate distributions====

Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.

With finite samples, approximation results measure how close a limiting distribution approaches the statistic's sample distribution: For example, with 10,000 independent samples the normal distribution approximates (to two digits of accuracy) the distribution of the sample mean for many population distributions, by the Berry–Esseen theorem.
Yet for many practical purposes, the normal approximation provides a good approximation to the sample-mean's distribution when there are 10 (or more) independent samples, according to simulation studies and statisticians' experience. Following Kolmogorov's work in the 1950s, advanced statistics uses approximation theory and functional analysis to quantify the error of approximation. In this approach, the metric geometry of probability distributions is studied; this approach quantifies approximation error with, for example, the Kullback–Leibler divergence, Bregman divergence, and the Hellinger distance.

With indefinitely large samples, limiting results like the central limit theorem describe the sample statistic's limiting distribution if one exists. Limiting results are not statements about finite samples, and indeed are irrelevant to finite samples. However, the asymptotic theory of limiting distributions is often invoked for work with finite samples. For example, limiting results are often invoked to justify the generalized method of moments and the use of generalized estimating equations, which are popular in econometrics and biostatistics. The magnitude of the difference between the limiting distribution and the true distribution (formally, the 'error' of the approximation) can be assessed using simulation. The heuristic application of limiting results to finite samples is common practice in many applications, especially with low-dimensional models with log-concave likelihoods (such as with one-parameter exponential families).

===Randomization-based models===

For a given dataset that was produced by a randomization design, the randomization distribution of a statistic (under the null-hypothesis) is defined by evaluating the test statistic for all of the plans that could have been generated by the randomization design. In frequentist inference, the randomization allows inferences to be based on the randomization distribution rather than a subjective model, and this is important especially in survey sampling and design of experiments. Statistical inference from randomized studies is also more straightforward than many other situations. In Bayesian inference, randomization is also of importance: in survey sampling, use of sampling without replacement ensures the exchangeability of the sample with the population; in randomized experiments, randomization warrants a missing at random assumption for covariate information.

Objective randomization allows properly inductive procedures. Many statisticians prefer randomization-based analysis of data that was generated by well-defined randomization procedures. (However, it is true that in fields of science with developed theoretical knowledge and experimental control, randomized experiments may increase the costs of experimentation without improving the quality of inferences.)
Similarly, results from randomized experiments are recommended by leading statistical authorities as allowing inferences with greater reliability than do observational studies of the same phenomena. However, a good observational study may be better than a bad randomized experiment.

The statistical analysis of a randomized experiment may be based on the randomization scheme stated in the experimental protocol and does not need a subjective model.

However, at any time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples. In some cases, such randomized studies are uneconomical or unethical.

==== Model-based analysis of randomized experiments ====
It is standard practice to refer to a statistical model, e.g., a linear or logistic models, when analyzing data from randomized experiments. However, the randomization scheme guides the choice of a statistical model. It is not possible to choose an appropriate model without knowing the randomization scheme. Seriously misleading results can be obtained analyzing data from randomized experiments while ignoring the experimental protocol; common mistakes include forgetting the blocking used in an experiment and confusing repeated measurements on the same experimental unit with independent replicates of the treatment applied to different experimental units.

==== Model-free randomization inference ====
Model-free techniques provide a complement to model-based methods, which employ reductionist strategies of reality-simplification. The former combine, evolve, ensemble and train algorithms dynamically adapting to the contextual affinities of a process and learning the intrinsic characteristics of the observations.

For example, model-free simple linear regression is based either on
* a ''random design'', where the pairs of observations <math>(X_1,Y_1), (X_2,Y_2), \cdots , (X_n,Y_n)</math> are independent and identically distributed (iid), or
* a ''deterministic design'', where the variables <math>X_1, X_2, \cdots, X_n</math> are deterministic, but the corresponding response variables <math>Y_1,Y_2, \cdots, Y_n</math> are random and independent with a common conditional distribution, i.e., <math>P\left (Y_j \leq y | X_j =x\right ) = D_x(y)</math>, which is independent of the index <math>j</math>.

In either case, the model-free randomization inference for features of the common conditional distribution <math>D_x(.)</math> relies on some regularity conditions, e.g. functional smoothness. For instance, model-free randomization inference for the population feature ''conditional mean'', <math>\mu(x)=E(Y | X = x)</math>, can be consistently estimated via local averaging or local polynomial fitting, under the assumption that <math>\mu(x)</math> is smooth. Also, relying on asymptotic normality or resampling, we can construct confidence intervals for the population feature, in this case, the ''conditional mean'', <math>\mu(x)</math>.

== Paradigms for inference ==
Different schools of statistical inference have become established. These schools—or "paradigms"—are not mutually exclusive, and methods that work well under one paradigm often have attractive interpretations under other paradigms.

Bandyopadhyay & Forster describe four paradigms: "(i) classical statistics or error statistics, (ii) Bayesian statistics, (iii) likelihood-based statistics, and (iv) the Akaikean-Information Criterion-based statistics". The classical (or frequentist) paradigm, the Bayesian paradigm, the likelihoodist paradigm, and the AIC-based paradigm are summarized below.

=== Frequentist inference ===

This paradigm calibrates the plausibility of propositions by considering (notional) repeated sampling of a population distribution to produce datasets similar to the one at hand. By considering the dataset's characteristics under repeated sampling, the frequentist properties of a statistical proposition can be quantified—although in practice this quantification may be challenging.

==== Examples of frequentist inference ====
* ''p''-value
* Confidence interval
* Null hypothesis significance testing

==== Frequentist inference, objectivity, and decision theory ====
One interpretation of frequentist inference (or classical inference) is that it is applicable only in terms of frequency probability; that is, in terms of repeated sampling from a population. However, the approach of Neyman develops these procedures in terms of pre-experiment probabilities. That is, before undertaking an experiment, one decides on a rule for coming to a conclusion such that the probability of being correct is controlled in a suitable way: such a probability need not have a frequentist or repeated sampling interpretation. In contrast, Bayesian inference works in terms of conditional probabilities (i.e. probabilities conditional on the observed data), compared to the marginal (but conditioned on unknown parameters) probabilities used in the frequentist approach.

The frequentist procedures of significance testing and confidence intervals can be constructed without regard to utility functions. However, some elements of frequentist statistics, such as statistical decision theory, do incorporate utility functions. In particular, frequentist developments of optimal inference (such as minimum-variance unbiased estimators, or uniformly most powerful testing) make use of loss functions, which play the role of (negative) utility functions. Loss functions need not be explicitly stated for statistical theorists to prove that a statistical procedure has an optimality property. However, loss-functions are often useful for stating optimality properties: for example, median-unbiased estimators are optimal under absolute value loss functions, in that they minimize expected loss, and least squares estimators are optimal under squared error loss functions, in that they minimize expected loss.

While statisticians using frequentist inference must choose for themselves the parameters of interest, and the estimators/test statistic to be used, the absence of obviously explicit utilities and prior distributions has helped frequentist procedures to become widely viewed as 'objective'.

===Bayesian inference===

The Bayesian calculus describes degrees of belief using the 'language' of probability; beliefs are positive, integrate into one, and obey probability axioms. Bayesian inference uses the available posterior beliefs as the basis for making statistical propositions. There are several different justifications for using the Bayesian approach.

====Examples of Bayesian inference====

* Credible interval for interval estimation
* Bayes factors for model comparison

====Bayesian inference, subjectivity and decision theory====

Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior. For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way. While a user's utility function need not be stated for this sort of inference, these summaries do all depend (to some extent) on stated prior beliefs, and are generally viewed as subjective conclusions. (Methods of prior construction which do not require external input have been proposed but not yet fully developed.)

Formally, Bayesian inference is calibrated with reference to an explicitly stated utility, or loss function; the 'Bayes rule' is the one which maximizes expected utility, averaged over the posterior uncertainty. Formal Bayesian inference therefore automatically provides optimal decisions in a decision theoretic sense. Given assumptions, data and utility, Bayesian inference can be made for essentially any problem, although not every statistical inference need have a Bayesian interpretation. Analyses which are not formally Bayesian can be (logically) incoherent; a feature of Bayesian procedures which use proper priors (i.e. those integrable to one) is that they are guaranteed to be coherent. Some advocates of Bayesian inference assert that inference ''must'' take place in this decision-theoretic framework, and that Bayesian inference should not conclude with the evaluation and summarization of posterior beliefs.

===Likelihood-based inference===

Likelihoodism approaches statistics by using the likelihood function. Some likelihoodists reject inference, considering statistics as only computing support from evidence. Others, however, propose inference based on the likelihood function, of which the best-known is maximum likelihood estimation.

===AIC-based inference===

The ''Akaike information criterion'' (AIC) is an estimator of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.

AIC is founded on information theory: it offers an estimate of the relative information lost when a given model is used to represent the process that generated the data. (In doing so, it deals with the trade-off between the goodness of fit of the model and the simplicity of the model.)

===Other paradigms for inference===

====Minimum description length====

The minimum description length (MDL) principle has been developed from ideas in information theory and the theory of Kolmogorov complexity. The (MDL) principle selects statistical models that maximally compress the data; inference proceeds without assuming counterfactual or non-falsifiable "data-generating mechanisms" or probability models for the data, as might be done in frequentist or Bayesian approaches.

However, if a "data generating mechanism" does exist in reality, then according to Shannon's source coding theorem it provides the MDL description of the data, on average and asymptotically. In minimizing description length (or descriptive complexity), MDL estimation is similar to maximum likelihood estimation and maximum a posteriori estimation (using maximum-entropy Bayesian priors). However, MDL avoids assuming that the underlying probability model is known; the MDL principle can also be applied without assumptions that e.g. the data arose from independent sampling.

The MDL principle has been applied in communication-coding theory in information theory, in linear regression, and in data mining.

The evaluation of MDL-based inferential procedures often uses techniques or criteria from computational complexity theory.

====Fiducial inference====
Fiducial inference was an approach to statistical inference based on fiducial probability, also known as a "fiducial distribution". In subsequent work, this approach has been called ill-defined, extremely limited in applicability, and even fallacious. However this argument is the same as that which shows that a so-called confidence distribution is not a valid probability distribution and, since this has not invalidated the application of confidence intervals, it does not necessarily invalidate conclusions drawn from fiducial arguments. An attempt was made to reinterpret the early work of Fisher's fiducial argument as a special case of an inference theory using Upper and lower probabilities.

====Structural inference====

Developing ideas of Fisher and of Pitman from 1938 to 1939, George A. Barnard developed "structural inference" or "pivotal inference", an approach using invariant probabilities on group families. Barnard reformulated the arguments behind fiducial inference on a restricted class of models on which "fiducial" procedures would be well-defined and useful. Donald A. S. Fraser developed a general theory for structural inference based on group theory and applied this to linear models. The theory formulated by Fraser has close links to decision theory and Bayesian statistics and can provide optimal frequentist decision rules if they exist.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Statistical_inference Statistical inference, Wikipedia] under a CC BY-SA license

Mathematical & Statistical Reasoning

2022-02-06T03:48:44Z

Khanh: Created page with "'''Statistical inference''' is the process of using data analysis to infer properties of an underlying distribution of probability. '''Inferential statistical analysis''' infe..."

'''Statistical inference''' is the process of using data analysis to infer properties of an underlying distribution of probability. '''Inferential statistical analysis''' infers properties of a population, for example by '''testing hypotheses''' and deriving estimates. It is assumed that the observed data set is sampled from a larger population.

'''Inferential statistics''' can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. In machine learning, the term ''inference'' is sometimes used instead to mean "make a prediction, by evaluating an already trained model"; in this context inferring properties of the model is referred to as ''training'' or ''learning'' (rather than ''inference''), and using a model for prediction is referred to as ''inference'' (instead of ''prediction''); see also predictive inference.

== Introduction ==
Statistical inference makes propositions about a population, using data drawn from the population with some form of sampling. Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) selecting a statistical model of the process that generates the data and (second) deducing propositions from the model.

Konishi & Kitagawa state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling". Relatedly, Sir David Cox has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".

The conclusion of a statistical inference is a statistical proposition. Some common forms of statistical proposition are the following:
* a point estimate, i.e. a particular value that best approximates some parameter of interest;
* an interval estimate, e.g. a confidence interval (or set estimate), i.e. an interval constructed using a dataset drawn from a population so that, under repeated sampling of such datasets, such intervals would contain the true parameter value with the probability at the stated confidence level;
* a credible interval, i.e. a set of values containing, for example, 95% of posterior belief;
* rejection of a hypothesis;
* clustering or classification of data points into groups.

== Models and assumptions ==

Any statistical inference requires some assumptions. A '''statistical model''' is a set of assumptions concerning the generation of the observed data and similar data. Descriptions of statistical models usually emphasize the role of population quantities of interest, about which we wish to draw inference. Descriptive statistics are typically used as a preliminary step before more formal inferences are drawn.

=== Degree of models/assumptions ===
Statisticians distinguish between three levels of modeling assumptions;
* '''Fully parametric''': The probability distributions describing the data-generation process are assumed to be fully described by a family of probability distributions involving only a finite number of unknown parameters. For example, one may assume that the distribution of population values is truly Normal, with unknown mean and variance, and that datasets are generated by 'simple' random sampling. The family of generalized linear models is a widely used and flexible class of parametric models.
* '''Non-parametric''': The assumptions made about the process generating the data are much less than in parametric statistics and may be minimal. For example, every continuous probability distribution has a median, which may be estimated using the sample median or the Hodges–Lehmann–Sen estimator, which has good properties when the data arise from simple random sampling.
* '''Semi-parametric''': This term typically implies assumptions 'in between' fully and non-parametric approaches. For example, one may assume that a population distribution has a finite mean. Furthermore, one may assume that the mean response level in the population depends in a truly linear manner on some covariate (a parametric assumption) but not make any parametric assumption describing the variance around that mean (i.e. about the presence or possible form of any heteroscedasticity). More generally, semi-parametric models can often be separated into 'structural' and 'random variation' components. One component is treated parametrically and the other non-parametrically. The well-known Cox model is a set of semi-parametric assumptions.

=== Importance of valid models/assumptions ===

Whatever level of assumption is made, correctly calibrated inference, in general, requires these assumptions to be correct; i.e. that the data-generating mechanisms really have been correctly specified.

Incorrect assumptions of ' simple' random sampling can invalidate statistical inference. More complex semi- and fully parametric assumptions are also cause for concern. For example, incorrectly assuming the Cox model can in some cases lead to faulty conclusions. Incorrect assumptions of Normality in the population also invalidates some forms of regression-based inference. The use of '''any''' parametric model is viewed skeptically by most experts in sampling human populations: "most sampling statisticians, when they deal with confidence intervals at all, limit themselves to statements about [estimators] based on very large samples, where the central limit theorem ensures that these [estimators] will have distributions that are nearly normal." In particular, a normal distribution "would be a totally unrealistic and catastrophically unwise assumption to make if we were dealing with any kind of economic population." Here, the central limit theorem states that the distribution of the sample mean "for very large samples" is approximately normally distributed, if the distribution is not heavy-tailed.

====Approximate distributions====

Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.

With finite samples, approximation results measure how close a limiting distribution approaches the statistic's sample distribution: For example, with 10,000 independent samples the [[normal distribution]] approximates (to two digits of accuracy) the distribution of the [[sample mean]] for many population distributions, by the Berry–Esseen theorem.
Yet for many practical purposes, the normal approximation provides a good approximation to the sample-mean's distribution when there are 10 (or more) independent samples, according to simulation studies and statisticians' experience. Following Kolmogorov's work in the 1950s, advanced statistics uses [[approximation theory]] and [[functional analysis]] to quantify the error of approximation. In this approach, the [[metric geometry]] of [[probability distribution]]s is studied; this approach quantifies approximation error with, for example, the [[Kullback–Leibler divergence]], [[Bregman divergence]], and the [[Hellinger distance]].

With indefinitely large samples, [[asymptotic theory (statistics)|limiting results]] like the [[central limit theorem]] describe the sample statistic's limiting distribution if one exists. Limiting results are not statements about finite samples, and indeed are irrelevant to finite samples. However, the asymptotic theory of limiting distributions is often invoked for work with finite samples. For example, limiting results are often invoked to justify the [[generalized method of moments]] and the use of [[generalized estimating equation]]s, which are popular in [[econometrics]] and [[biostatistics]]. The magnitude of the difference between the limiting distribution and the true distribution (formally, the 'error' of the approximation) can be assessed using simulation. The heuristic application of limiting results to finite samples is common practice in many applications, especially with low-dimensional [[statistical model|models]] with [[logarithmically concave function|log-concave]] [[likelihood function|likelihood]]s (such as with one-parameter [[exponential families]]).

===Randomization-based models===

For a given dataset that was produced by a randomization design, the randomization distribution of a statistic (under the null-hypothesis) is defined by evaluating the test statistic for all of the plans that could have been generated by the randomization design. In frequentist inference, the randomization allows inferences to be based on the randomization distribution rather than a subjective model, and this is important especially in survey sampling and design of experiments.<ref>[[Jerzy Neyman|Neyman, J.]](1934) "On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection", ''[[Journal of the Royal Statistical Society]]'', 97 (4), 557–625 {{jstor|2342192}}</ref><ref name="Hinkelmann and Kempthorne">Hinkelmann and Kempthorne(2008) {{page needed|date=June 2011}}</ref> Statistical inference from randomized studies is also more straightforward than many other situations.<ref>ASA Guidelines for the first course in statistics for non-statisticians. (available at the ASA website)</ref><ref>[[David A. Freedman]] et alia's ''Statistics''.</ref><ref>Moore et al. (2015).</ref> In [[Bayesian inference]], randomization is also of importance: in [[survey sampling]], use of [[sampling without replacement]] ensures the [[exchangeability]] of the sample with the population; in randomized experiments, randomization warrants a [[missing at random]] assumption for [[covariate]] information.

Objective randomization allows properly inductive procedures. Many statisticians prefer randomization-based analysis of data that was generated by well-defined randomization procedures. (However, it is true that in fields of science with developed theoretical knowledge and experimental control, randomized experiments may increase the costs of experimentation without improving the quality of inferences.)
Similarly, results from randomized experiments are recommended by leading statistical authorities as allowing inferences with greater reliability than do observational studies of the same phenomena. However, a good observational study may be better than a bad randomized experiment.

The statistical analysis of a randomized experiment may be based on the randomization scheme stated in the experimental protocol and does not need a subjective model.

However, at any time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples. In some cases, such randomized studies are uneconomical or unethical.

==== Model-based analysis of randomized experiments ====
It is standard practice to refer to a statistical model, e.g., a linear or logistic models, when analyzing data from randomized experiments. However, the randomization scheme guides the choice of a statistical model. It is not possible to choose an appropriate model without knowing the randomization scheme. Seriously misleading results can be obtained analyzing data from randomized experiments while ignoring the experimental protocol; common mistakes include forgetting the blocking used in an experiment and confusing repeated measurements on the same experimental unit with independent replicates of the treatment applied to different experimental units.

==== Model-free randomization inference ====
Model-free techniques provide a complement to model-based methods, which employ reductionist strategies of reality-simplification. The former combine, evolve, ensemble and train algorithms dynamically adapting to the contextual affinities of a process and learning the intrinsic characteristics of the observations.

For example, model-free simple linear regression is based either on
* a ''random design'', where the pairs of observations <math>(X_1,Y_1), (X_2,Y_2), \cdots , (X_n,Y_n)</math> are independent and identically distributed (iid), or
* a ''deterministic design'', where the variables <math>X_1, X_2, \cdots, X_n</math> are deterministic, but the corresponding response variables <math>Y_1,Y_2, \cdots, Y_n</math> are random and independent with a common conditional distribution, i.e., <math>P\left (Y_j \leq y | X_j =x\right ) = D_x(y)</math>, which is independent of the index <math>j</math>.

In either case, the model-free randomization inference for features of the common conditional distribution <math>D_x(.)</math> relies on some regularity conditions, e.g. functional smoothness. For instance, model-free randomization inference for the population feature ''conditional mean'', <math>\mu(x)=E(Y | X = x)</math>, can be consistently estimated via local averaging or local polynomial fitting, under the assumption that <math>\mu(x)</math> is smooth. Also, relying on asymptotic normality or resampling, we can construct confidence intervals for the population feature, in this case, the ''conditional mean'', <math>\mu(x)</math>.

== Paradigms for inference ==
Different schools of statistical inference have become established. These schools—or "paradigms"—are not mutually exclusive, and methods that work well under one paradigm often have attractive interpretations under other paradigms.

Bandyopadhyay & Forster describe four paradigms: "(i) classical statistics or error statistics, (ii) Bayesian statistics, (iii) likelihood-based statistics, and (iv) the Akaikean-Information Criterion-based statistics". The classical (or frequentist) paradigm, the Bayesian paradigm, the likelihoodist paradigm, and the AIC-based paradigm are summarized below.

=== Frequentist inference ===

This paradigm calibrates the plausibility of propositions by considering (notional) repeated sampling of a population distribution to produce datasets similar to the one at hand. By considering the dataset's characteristics under repeated sampling, the frequentist properties of a statistical proposition can be quantified—although in practice this quantification may be challenging.

==== Examples of frequentist inference ====
* ''p''-value
* Confidence interval
* Null hypothesis significance testing

==== Frequentist inference, objectivity, and decision theory ====
One interpretation of frequentist inference (or classical inference) is that it is applicable only in terms of frequency probability; that is, in terms of repeated sampling from a population. However, the approach of Neyman develops these procedures in terms of pre-experiment probabilities. That is, before undertaking an experiment, one decides on a rule for coming to a conclusion such that the probability of being correct is controlled in a suitable way: such a probability need not have a frequentist or repeated sampling interpretation. In contrast, Bayesian inference works in terms of conditional probabilities (i.e. probabilities conditional on the observed data), compared to the marginal (but conditioned on unknown parameters) probabilities used in the frequentist approach.

The frequentist procedures of significance testing and confidence intervals can be constructed without regard to utility functions. However, some elements of frequentist statistics, such as statistical decision theory, do incorporate utility functions. In particular, frequentist developments of optimal inference (such as minimum-variance unbiased estimators, or uniformly most powerful testing) make use of loss functions, which play the role of (negative) utility functions. Loss functions need not be explicitly stated for statistical theorists to prove that a statistical procedure has an optimality property. However, loss-functions are often useful for stating optimality properties: for example, median-unbiased estimators are optimal under absolute value loss functions, in that they minimize expected loss, and least squares estimators are optimal under squared error loss functions, in that they minimize expected loss.

While statisticians using frequentist inference must choose for themselves the parameters of interest, and the estimators/test statistic to be used, the absence of obviously explicit utilities and prior distributions has helped frequentist procedures to become widely viewed as 'objective'.

===Bayesian inference===

The Bayesian calculus describes degrees of belief using the 'language' of probability; beliefs are positive, integrate into one, and obey probability axioms. Bayesian inference uses the available posterior beliefs as the basis for making statistical propositions. There are several different justifications for using the Bayesian approach.

====Examples of Bayesian inference====

* Credible interval for interval estimation
* Bayes factors for model comparison

====Bayesian inference, subjectivity and decision theory====

Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior. For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way. While a user's utility function need not be stated for this sort of inference, these summaries do all depend (to some extent) on stated prior beliefs, and are generally viewed as subjective conclusions. (Methods of prior construction which do not require external input have been proposed but not yet fully developed.)

Formally, Bayesian inference is calibrated with reference to an explicitly stated utility, or loss function; the 'Bayes rule' is the one which maximizes expected utility, averaged over the posterior uncertainty. Formal Bayesian inference therefore automatically provides optimal decisions in a decision theoretic sense. Given assumptions, data and utility, Bayesian inference can be made for essentially any problem, although not every statistical inference need have a Bayesian interpretation. Analyses which are not formally Bayesian can be (logically) incoherent; a feature of Bayesian procedures which use proper priors (i.e. those integrable to one) is that they are guaranteed to be coherent. Some advocates of Bayesian inference assert that inference ''must'' take place in this decision-theoretic framework, and that Bayesian inference should not conclude with the evaluation and summarization of posterior beliefs.

===Likelihood-based inference===

Likelihoodism approaches statistics by using the likelihood function. Some likelihoodists reject inference, considering statistics as only computing support from evidence. Others, however, propose inference based on the likelihood function, of which the best-known is maximum likelihood estimation.

===AIC-based inference===

The ''Akaike information criterion'' (AIC) is an estimator of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.

AIC is founded on information theory: it offers an estimate of the relative information lost when a given model is used to represent the process that generated the data. (In doing so, it deals with the trade-off between the goodness of fit of the model and the simplicity of the model.)

===Other paradigms for inference===

====Minimum description length====

The minimum description length (MDL) principle has been developed from ideas in information theory and the theory of Kolmogorov complexity. The (MDL) principle selects statistical models that maximally compress the data; inference proceeds without assuming counterfactual or non-falsifiable "data-generating mechanisms" or probability models for the data, as might be done in frequentist or Bayesian approaches.

However, if a "data generating mechanism" does exist in reality, then according to Shannon's source coding theorem it provides the MDL description of the data, on average and asymptotically. In minimizing description length (or descriptive complexity), MDL estimation is similar to maximum likelihood estimation and maximum a posteriori estimation (using maximum-entropy Bayesian priors). However, MDL avoids assuming that the underlying probability model is known; the MDL principle can also be applied without assumptions that e.g. the data arose from independent sampling.

The MDL principle has been applied in communication-coding theory in information theory, in linear regression, and in data mining.

The evaluation of MDL-based inferential procedures often uses techniques or criteria from computational complexity theory.

====Fiducial inference====
Fiducial inference was an approach to statistical inference based on fiducial probability, also known as a "fiducial distribution". In subsequent work, this approach has been called ill-defined, extremely limited in applicability, and even fallacious. However this argument is the same as that which shows that a so-called confidence distribution is not a valid probability distribution and, since this has not invalidated the application of confidence intervals, it does not necessarily invalidate conclusions drawn from fiducial arguments. An attempt was made to reinterpret the early work of Fisher's fiducial argument as a special case of an inference theory using Upper and lower probabilities.

====Structural inference====

Developing ideas of Fisher and of Pitman from 1938 to 1939, George A. Barnard developed "structural inference" or "pivotal inference", an approach using invariant probabilities on group families. Barnard reformulated the arguments behind fiducial inference on a restricted class of models on which "fiducial" procedures would be well-defined and useful. Donald A. S. Fraser developed a general theory for structural inference based on group theory and applied this to linear models. The theory formulated by Fraser has close links to decision theory and Bayesian statistics and can provide optimal frequentist decision rules if they exist.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Statistical_inference Statistical inference, Wikipedia] under a CC BY-SA license

Number Systems, Base 10, 5 and 2

2022-02-06T03:15:23Z

Khanh: /* Main numeral systems */

[[File:Numeral Systems of the World.svg|264px|thumb|right|Numbers written in different numeral systems.]]

A '''numeral system''' (or '''system of numeration''') is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number ''eleven'' in the decimal numeral system (used in common life), the number ''three'' in the binary numeral system (used in computers), and the number ''two'' in the unary numeral system (e.g. used in tallying scores).

The number the numeral represents is called its value. Not all number systems can represent all numbers that are considered in the modern days; for example, Roman numerals have no zero.

Ideally, a numeral system will:
*Represent a useful set of numbers (e.g. all integers, or rational numbers)
*Give every number represented a unique representation (or at least a standard representation)
*Reflect the algebraic and arithmetic structure of the numbers.

For example, the usual decimal representation gives every nonzero natural number a unique representation as a finite sequence of digits, beginning with a non-zero digit.

Numeral systems are sometimes called ''number systems'', but that name is ambiguous, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of ''p''-adic numbers, etc. Such systems are, however, not the topic of this article.

==Main numeral systems==
The most commonly used system of numerals is decimal. Indian mathematicians are credited with developing the integer version, the Hindu–Arabic numeral system. Aryabhata of Kusumapura developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. Middle-Eastern mathematicians extended the system to include negative powers of 10 (fractions), as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952–953, and the decimal point notation was introduced by Sind ibn Ali, who also wrote the earliest treatise on Arabic numerals. The Hindu-Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called Arabic numerals, as they learned them from the Arabs.

The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the number seven would be represented by / / / / / / / . Tally marks represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is commonly used in data compression, expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.

The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as +++ / / / / and the number 123 as + − − / / / without any need for zero. This is called sign-value notation. The ancient Egyptian numeral system was of this type, and the Roman numeral system was a modification of this idea.

More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of the alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, one could then write C+ D/ for the number 304. This system is used when writing Chinese numerals and other East Asian numerals based on Chinese. The number system of the English language is of this type ("three hundred [and] four"), as are those of other spoken languages, regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for instance 79 in French is ''soixante dix-neuf'' (60 + 10 + 9) and in Welsh is ''pedwar ar bymtheg a thrigain'' (4 + (5 + 10) + (3 × 20)) or (somewhat archaic) ''pedwar ugain namyn un'' (4 × 20 − 1). In English, one could say "four score less one", as in the famous Gettysburg Address representing "87 years ago" as "four score and seven years ago".

More elegant is a ''positional system'', also known as place-value notation. Again working in base 10, ten different digits 0, ..., 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1 or more precisely 3×102 + 0×101 + 4×100. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base 10).

The positional decimal system is presently universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers.

In computers, the main numeral systems are based on the positional system in base 2 (binary numeral system), with two binary digits, 0 and 1. Positional systems obtained by grouping binary digits by three (octal numeral system) or four (hexadecimal numeral system) are commonly used. For very large integers, bases 232 or 264 (grouping binary digits by 32 or 64, the length of the machine word) are used, as, for example, in GMP.

In certain biological systems, the unary coding system is employed. Unary numerals used in the neural circuits responsible for birdsong production. The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC (high vocal center). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.

The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the geometric numerals (1, 10, 100, 1000, 10000 ...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the Ionic system), and a positional system does not need geometric numerals because they are made by position. However, the spoken language uses ''both'' arithmetic and geometric numerals.

In certain areas of computer science, a modified base ''k'' positional system is used, called bijective numeration, with digits 1, 2, ..., ''k'' (''k'' ≥ 1), and zero being represented by an empty string. This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-''k'' numeration is also called ''k''-adic notation, not to be confused with ''p''-adic numbers. Bijective base 1 is the same as unary.

==Positional systems in detail==
In a positional base ''b'' numeral system (with ''b'' a natural number greater than 1 known as the radix), ''b'' basic symbols (or digits) corresponding to the first ''b'' natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by ''b''.

For example, in the decimal system (base 10), the numeral 4327 means {{math|('''4'''×103) + ('''3'''×102) + ('''2'''×101) + ('''7'''×100)}}, noting that 100 = 1.

In general, if ''b'' is the base, one writes a number in the numeral system of base ''b'' by expressing it in the form {{math|''a''''n''''b''''n'' + ''a''''n'' − 1''b''''n'' − 1 + ''a''''n'' − 2''b''''n'' − 2 + ... + ''a''0''b''0}} and writing the enumerated digits {{math|''a''''n''''a''''n'' − 1''a''''n'' − 2 ... ''a''0}} in descending order. The digits are natural numbers between 0 and {{math|''b'' − 1}}, inclusive.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: numberbase. Unless specified by context, numbers without subscript are considered to be decimal.

By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base 2 numeral 10.11 denotes 1×21 + 0×20 + 1×2−1 + 1×2−2 = 2.75.

In general, numbers in the base ''b'' system are of the form:

:<math>
(a_na_{n-1}\cdots a_1a_0.c_1 c_2 c_3\cdots)_b =
\sum_{k=0}^n a_kb^k + \sum_{k=1}^\infty c_kb^{-k}.
</math>

The numbers ''b''''k'' and ''b''−''k'' are the weights of the corresponding digits. The position ''k'' is the logarithm of the corresponding weight ''w'', that is <math>k = \log_{b} w = \log_{b} b^k</math>. The highest used position is close to the order of magnitude of the number.

The number of tally marks required in the unary numeral system for ''describing the weight'' would have been '''w'''. In the positional system, the number of digits required to describe it is only <math>k + 1 = \log_{b} w + 1</math>, for ''k'' ≥ 0. For example, to describe the weight 1000 then four digits are needed because <math>\log_{10} 1000 + 1 = 3 + 1</math>. The number of digits required to ''describe the position'' is <math>\log_b k + 1 = \log_b \log_b w + 1</math> (in positions 1, 10, 100,... only for simplicity in the decimal example).

:<math>\begin{array}{l|rrrrrrr}
\text{Position}
& 3
& 2
& 1
& 0
& -1
& -2
& \cdots
\\
\hline
\text{Weight}
& b^3 & b^2 & b^1 & b^0 & b^{-1} & b^{-2} & \cdots
\\
\text{Digit}
& a_3 & a_2 & a_1 & a_0 & c_1 & c_2 & \cdots
\\
\hline
\text{Decimal example weight}
& 1000 & 100 & 10 & 1 & 0.1 & 0.01 & \cdots
\\
\text{Decimal example digit}
& 4 & 3 & 2 & 7 & 0 & 0 & \cdots
\end{array}
</math>

A number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.310 = 0.0100110011001...2). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926...10 can be written as the aperiodic 11.001001000011111...2.

Putting overscores, <math> \overline{n}</math>, or dots, ''ṅ'', above the common digits is a convention used to represent repeating rational expansions. Thus:
:14/11 = 1.272727272727... = <math> 1. \overline{27}</math>   or   321.3217878787878... = <math> 321.321 \overline{78}</math>.

If ''b'' = ''p'' is a prime number, one can define base-''p'' numerals whose expansion to the left never stops; these are called the ''p''-adic numbers.

==Generalized variable-length integers==
More general is using a mixed radix notation (here written little-endian) like <math>a_0 a_1 a_2</math> for <math>a_0 + a_1 b_1 + a_2 b_1 b_2</math>, etc.

This is used in Punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a–z and 0–9, representing 0–25 and 26–35 respectively. There are also so-called threshold values (<math>t_0, t_1, ...</math>) which are fixed for every position in the number. A digit <math>a_i</math> (in a given position in the number) that is lower than its corresponding threshold value <math>t_i</math> means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next number.

For example, if the threshold value for the first digit is ''b'' (i.e. 1) then ''a'' (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit, first-digit range is only b–9 (i.e. 1–35), therefore the weight ''b''1 is 35 instead of 36. More generally, if ''tn'' is the threshold for the ''n''-th digit, it is easy to show that <math>b_{n+1}=36-t_n</math>.
Suppose the threshold values for the second and third digits are ''c'' (i.e. 2), then the second-digit range is a–b (i.e. 0–1) with the second digit being most significant, while the range is c–9 (i.e. 2–35) in the presence of a third digit. Generally, for any ''n'', the weight of the (''n''+1)-th digit is the weight of the previous one times (36 − threshold of the ''n''-th digit). So the weight of the second symbol is <math>36 - t_0 = 35</math>. And the weight of the third symbol is <math>35 * (36 - t_1) = 35*34 = 1190</math>.

So we have the following sequence of the numbers with at most 3 digits:

''a'' (0), ''ba'' (1), ''ca'' (2), ..., ''9a'' (35), ''bb'' (36), ''cb'' (37), ..., ''9b'' (70), ''bca'' (71), ..., ''99a'' (1260), ''bcb'' (1261), ..., ''99b'' (2450).

Unlike a regular n-based numeral system, there are numbers like ''9b'' where ''9'' and ''b'' each represent 35; yet the representation is unique because ''ac'' and ''aca'' are not allowed – the first ''a'' would terminate each of the se numbers.

The flexibility in choosing threshold values allows optimization for number of digits depending on the frequency of occurrence of numbers of various sizes.

The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are non-zero.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Numeral_system Numeral system, Wikipedia] under a CC BY-SA license

Number Systems, Base 10, 5 and 2

2022-02-06T02:48:05Z

Khanh:

[[File:Numeral Systems of the World.svg|264px|thumb|right|Numbers written in different numeral systems.]]

A '''numeral system''' (or '''system of numeration''') is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number ''eleven'' in the decimal numeral system (used in common life), the number ''three'' in the binary numeral system (used in computers), and the number ''two'' in the unary numeral system (e.g. used in tallying scores).

The number the numeral represents is called its value. Not all number systems can represent all numbers that are considered in the modern days; for example, Roman numerals have no zero.

Ideally, a numeral system will:
*Represent a useful set of numbers (e.g. all integers, or rational numbers)
*Give every number represented a unique representation (or at least a standard representation)
*Reflect the algebraic and arithmetic structure of the numbers.

For example, the usual decimal representation gives every nonzero natural number a unique representation as a finite sequence of digits, beginning with a non-zero digit.

Numeral systems are sometimes called ''number systems'', but that name is ambiguous, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of ''p''-adic numbers, etc. Such systems are, however, not the topic of this article.

==Main numeral systems==
The most commonly used system of numerals is decimal. Indian mathematicians are credited with developing the integer version, the Hindu–Arabic numeral system. Aryabhata of Kusumapura developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. Middle-Eastern mathematicians extended the system to include negative powers of 10 (fractions), as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952–953, and the decimal point notation was introduced by Sind ibn Ali, who also wrote the earliest treatise on Arabic numerals. The Hindu-Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called Arabic numerals, as they learned them from the Arabs.

The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the number seven would be represented by ///////. Tally marks represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is commonly used in data compression, expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.

The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as +++ //// and the number 123 as + − − /// without any need for zero. This is called sign-value notation. The ancient Egyptian numeral system was of this type, and the Roman numeral system was a modification of this idea.

More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of the alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, one could then write C+ D/ for the number 304. This system is used when writing Chinese numerals and other East Asian numerals based on Chinese. The number system of the English language is of this type ("three hundred [and] four"), as are those of other spoken languages, regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for instance 79 in French is ''soixante dix-neuf'' (60 + 10 + 9) and in Welsh is ''pedwar ar bymtheg a thrigain'' (4 + (5 + 10) + (3 × 20)) or (somewhat archaic) ''pedwar ugain namyn un'' (4 × 20 − 1). In English, one could say "four score less one", as in the famous Gettysburg Address representing "87 years ago" as "four score and seven years ago".

More elegant is a ''positional system'', also known as place-value notation. Again working in base 10, ten different digits 0, ..., 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1 or more precisely 3×102 + 0×101 + 4×100. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base 10).

The positional decimal system is presently universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers.

In computers, the main numeral systems are based on the positional system in base 2 (binary numeral system), with two binary digits, 0 and 1. Positional systems obtained by grouping binary digits by three (octal numeral system) or four (hexadecimal numeral system) are commonly used. For very large integers, bases 232 or 264 (grouping binary digits by 32 or 64, the length of the machine word) are used, as, for example, in GMP.

In certain biological systems, the unary coding system is employed. Unary numerals used in the neural circuits responsible for birdsong production. The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC (high vocal center). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.

The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the geometric numerals (1, 10, 100, 1000, 10000 ...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the Ionic system), and a positional system does not need geometric numerals because they are made by position. However, the spoken language uses ''both'' arithmetic and geometric numerals.

In certain areas of computer science, a modified base ''k'' positional system is used, called bijective numeration, with digits 1, 2, ..., ''k'' (''k'' ≥ 1), and zero being represented by an empty string. This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-''k'' numeration is also called ''k''-adic notation, not to be confused with ''p''-adic numbers. Bijective base 1 is the same as unary.

==Positional systems in detail==
In a positional base ''b'' numeral system (with ''b'' a natural number greater than 1 known as the radix), ''b'' basic symbols (or digits) corresponding to the first ''b'' natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by ''b''.

For example, in the decimal system (base 10), the numeral 4327 means {{math|('''4'''×103) + ('''3'''×102) + ('''2'''×101) + ('''7'''×100)}}, noting that 100 = 1.

In general, if ''b'' is the base, one writes a number in the numeral system of base ''b'' by expressing it in the form {{math|''a''''n''''b''''n'' + ''a''''n'' − 1''b''''n'' − 1 + ''a''''n'' − 2''b''''n'' − 2 + ... + ''a''0''b''0}} and writing the enumerated digits {{math|''a''''n''''a''''n'' − 1''a''''n'' − 2 ... ''a''0}} in descending order. The digits are natural numbers between 0 and {{math|''b'' − 1}}, inclusive.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: numberbase. Unless specified by context, numbers without subscript are considered to be decimal.

By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base 2 numeral 10.11 denotes 1×21 + 0×20 + 1×2−1 + 1×2−2 = 2.75.

In general, numbers in the base ''b'' system are of the form:

:<math>
(a_na_{n-1}\cdots a_1a_0.c_1 c_2 c_3\cdots)_b =
\sum_{k=0}^n a_kb^k + \sum_{k=1}^\infty c_kb^{-k}.
</math>

The numbers ''b''''k'' and ''b''−''k'' are the weights of the corresponding digits. The position ''k'' is the logarithm of the corresponding weight ''w'', that is <math>k = \log_{b} w = \log_{b} b^k</math>. The highest used position is close to the order of magnitude of the number.

The number of tally marks required in the unary numeral system for ''describing the weight'' would have been '''w'''. In the positional system, the number of digits required to describe it is only <math>k + 1 = \log_{b} w + 1</math>, for ''k'' ≥ 0. For example, to describe the weight 1000 then four digits are needed because <math>\log_{10} 1000 + 1 = 3 + 1</math>. The number of digits required to ''describe the position'' is <math>\log_b k + 1 = \log_b \log_b w + 1</math> (in positions 1, 10, 100,... only for simplicity in the decimal example).

:<math>\begin{array}{l|rrrrrrr}
\text{Position}
& 3
& 2
& 1
& 0
& -1
& -2
& \cdots
\\
\hline
\text{Weight}
& b^3 & b^2 & b^1 & b^0 & b^{-1} & b^{-2} & \cdots
\\
\text{Digit}
& a_3 & a_2 & a_1 & a_0 & c_1 & c_2 & \cdots
\\
\hline
\text{Decimal example weight}
& 1000 & 100 & 10 & 1 & 0.1 & 0.01 & \cdots
\\
\text{Decimal example digit}
& 4 & 3 & 2 & 7 & 0 & 0 & \cdots
\end{array}
</math>

A number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.310 = 0.0100110011001...2). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926...10 can be written as the aperiodic 11.001001000011111...2.

Putting overscores, <math> \overline{n}</math>, or dots, ''ṅ'', above the common digits is a convention used to represent repeating rational expansions. Thus:
:14/11 = 1.272727272727... = <math> 1. \overline{27}</math>   or   321.3217878787878... = <math> 321.321 \overline{78}</math>.

If ''b'' = ''p'' is a prime number, one can define base-''p'' numerals whose expansion to the left never stops; these are called the ''p''-adic numbers.

==Generalized variable-length integers==
More general is using a mixed radix notation (here written little-endian) like <math>a_0 a_1 a_2</math> for <math>a_0 + a_1 b_1 + a_2 b_1 b_2</math>, etc.

This is used in Punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a–z and 0–9, representing 0–25 and 26–35 respectively. There are also so-called threshold values (<math>t_0, t_1, ...</math>) which are fixed for every position in the number. A digit <math>a_i</math> (in a given position in the number) that is lower than its corresponding threshold value <math>t_i</math> means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next number.

For example, if the threshold value for the first digit is ''b'' (i.e. 1) then ''a'' (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit, first-digit range is only b–9 (i.e. 1–35), therefore the weight ''b''1 is 35 instead of 36. More generally, if ''tn'' is the threshold for the ''n''-th digit, it is easy to show that <math>b_{n+1}=36-t_n</math>.
Suppose the threshold values for the second and third digits are ''c'' (i.e. 2), then the second-digit range is a–b (i.e. 0–1) with the second digit being most significant, while the range is c–9 (i.e. 2–35) in the presence of a third digit. Generally, for any ''n'', the weight of the (''n''+1)-th digit is the weight of the previous one times (36 − threshold of the ''n''-th digit). So the weight of the second symbol is <math>36 - t_0 = 35</math>. And the weight of the third symbol is <math>35 * (36 - t_1) = 35*34 = 1190</math>.

So we have the following sequence of the numbers with at most 3 digits:

''a'' (0), ''ba'' (1), ''ca'' (2), ..., ''9a'' (35), ''bb'' (36), ''cb'' (37), ..., ''9b'' (70), ''bca'' (71), ..., ''99a'' (1260), ''bcb'' (1261), ..., ''99b'' (2450).

Unlike a regular n-based numeral system, there are numbers like ''9b'' where ''9'' and ''b'' each represent 35; yet the representation is unique because ''ac'' and ''aca'' are not allowed – the first ''a'' would terminate each of the se numbers.

The flexibility in choosing threshold values allows optimization for number of digits depending on the frequency of occurrence of numbers of various sizes.

The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are non-zero.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Numeral_system Numeral system, Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T19:28:59Z

Khanh: /* Licensing */

== Formal logical systems ==
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.

=== First-order logic ===
'''First-order logic''' is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.

Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.

Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The '''first incompleteness theorem''' states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The '''second incompleteness theorem''' states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

=== Other classical logics ===
Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''fixed-point logics''' that allow inductive definitions, like one writes for primitive recursive functions.

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

=== Nonclassical and modal logic ===
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

=== Algebraic logic ===
Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.

== Problem-solving strategies ==
Problem-solving strategies are the steps that one would use to find the problems that are in the way to getting to one's own goal. Some refer to this as the "problem-solving cycle".

In this cycle one will acknowledge, recognize the problem, define the problem, develop a strategy to fix the problem, organize the knowledge of the problem cycle, figure out the resources at the user's disposal, monitor one's progress, and evaluate the solution for accuracy. The reason it is called a cycle is that once one is completed with a problem another will usually pop up.

'''Insight''' is the sudden solution to a long-vexing '''problem''', a sudden recognition of a new idea, or a sudden understanding of a complex situation, an ''Aha!'' moment. Solutions found through '''insight''' are often more accurate than those found through step-by-step analysis. To solve more problems at a faster rate, insight is necessary for selecting productive moves at different stages of the problem-solving cycle. This problem-solving strategy pertains specifically to problems referred to as insight problem. Unlike Newell and Simon's formal definition of move problems, there has not been a generally agreed upon definition of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011).

Blanchard-Fields looks at problem solving from one of two facets. The first looking at those problems that only have one solution (like mathematical problems, or fact-based questions) which are grounded in psychometric intelligence. The other is socioemotional in nature and have answers that change constantly (like what's your favorite color or what you should get someone for Christmas).

The following techniques are usually called ''problem-solving strategies''
* Abstraction: solving the problem in a model of the system before applying it to the real system
* Analogy: using a solution that solves an analogous problem
* Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
* Critical thinking
* Divide and conquer: breaking down a large, complex problem into smaller, solvable problems
* Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
* Lateral thinking: approaching solutions indirectly and creatively
* Means-ends analysis: choosing an action at each step to move closer to the goal
* Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new
* Morphological analysis: assessing the output and interactions of an entire system
* Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
* Reduction: transforming the problem into another problem for which solutions exist
* Research: employing existing ideas or adapting existing solutions to similar problems
* Root cause analysis: identifying the cause of a problem
* Trial-and-error: testing possible solutions until the right one is found

== Multiple forms of representation ==
=== Concrete models===
====Base ten blocks====
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

====Pattern blocks====
[[File:Wooden pattern blocks dodecagon.JPG|thumb|One of the ways of making a dodecagon with pattern blocks]]
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.

Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.

Pattern blocks can also serve to provide students with an understanding of fractions. Because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.

Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

====Unifix® Cubes====
[[File:Linking cm cubes 2.JPG|thumb|Interlocking centimeter cubes]]
Unifix® Cubes are interlocking cubes that are just under 2 centimeters on each side. The cubes connect with each other from one side. Once connected, Unifix® Cubes can be turned to form a vertical Unifix® "tower," or horizontally to form a Unifix® "train".

Other interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities.

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence,
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue..
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.

Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume.

====Tiles====
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).

Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.

These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

====Number lines====
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

====Cuisenaire rods====
[[File:Cuisenaire ten.JPG|thumb|Cuisenaire rods used to illustrate the factors of ten]]
'''Cuisenaire rods''' are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods ''réglettes''.

According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."

;Use in mathematics teaching
The rods are used in teaching a variety of mathematical ideas, and with a wide age range of learners. Topics they are used for include:
* Counting, sequences, patterns and algebraic reasoning
* Addition and subtraction (additive reasoning)
* Multiplication and division (multiplicative reasoning)
* Fractions, ratio and proportion
* Modular arithmetic leading to group theory

=== Pictures ===
==== Pictogram ====
[[File:Titanic casualties.svg|thumb|300px|alt=table with boxes instead of numbers, the amounts and sizes of boxes represent amounts of people|A compound pictogram showing the breakdown of the survivors and deaths of the maiden voyage of the RMS Titanic by class and age/gender.]]

Pictograms are charts in which icons represent numbers to make it more interesting and easier to understand. A key is often included to indicate what each icon represents. All icons must be of the same size, but a fraction of an icon can be used to show the respective fraction of that amount.

For example, the following table:
{| class="wikitable"
! Day !! Letters sent
|-
| Monday || 10
|-
| Tuesday || 17
|-
| Wednesday || 29
|-
| Thursday || 41
|-
| Friday || 18
|}
can be graphed as follows:
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+
! Day !! Letters sent
|-
| Monday || [[File:Email Silk.svg|alt=one envelope]]
|-
| Tuesday || [[File:Email Silk.svg|alt=one envelope]] [[File:Image from the Silk icon theme by Mark James half left.svg|alt=and a half]]
|-
| Wednesday || [[File:Email Silk.svg|alt=three envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Thursday || [[File:Email Silk.svg|alt=four envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Friday || [[File:Email Silk.svg|alt=two envelopes]] [[File:Email Silk.svg|alt=]]
|}
Key: [[File:Email Silk.svg|alt=one envelope]] = 10 letters

As the values are rounded to the nearest 5 letters, the second icon on Tuesday is the left half of the original.

==== Tally marks ====
[[File:Strike symbol (49778996432).jpg |thumb|Tally marks on a chalkboard]]
'''Tally marks''', also called '''hash marks''', are a unary numeral system. They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded.

However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were also historically used for this purpose.

;Clustering
Tally marks are typically clustered in groups of five for legibility. The cluster size 5 has the advantages of (a) easy conversion into decimal for higher arithmetic operations and (b) avoiding error, as humans can far more easily correctly identify a cluster of 5 than one of 10.

<gallery>
File:Tally marks.svg|Tally marks high in most of Europe, Zimbabwe and Southern Africa, Australia, New Zealand and North America. In some variants, the diagonal/horizontal slash is used on its own when five or more units are added at once.
File:Tally marks 3.svg|Cultures using Chinese characters tally by forming the character 正, which consists of five strokes.
File:Tally marks 2.svg|Tally marks used in France, their former colonies, Argentina and Brazil. 1 to 5 and so on. These are most commonly used for registering scores in card games, like Truco
File:Dot and line tally marks.jpg|In the dot and line (or dot-dash) tally, dots represent counts from 1 to 4, lines 5 to 8, and diagonal lines 9 and 10. This method is commonly used in forestry and related fields.
</gallery>

=== Diagrams ===
A '''diagram''' is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram.

====Gallery of diagram types====

There are at least the following types of diagrams:

* Logical or conceptual diagrams, which take a collection of items and relationships between them, and express them by giving each item a 2D position, while the relationships are expressed as connections between the items or overlaps between the items, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Tree Example.png|Tree diagram
File:Neural network.svg|Network diagram
File:LampFlowchart.svg|Flowchart
File:Set intersection.svg|Venn diagram
File:Alphagraphen.png|Existential graph
</gallery>
|}

* Quantitative diagrams, which display a relationship between two variables that take either discrete or a continuous range of values; for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Histogram example.svg|Histogram
File:Graphtestone.svg|Bar graph
File:Zusammensetzung Shampoo.svg|Pie chart
File:Hyperbolic Cosine.svg|Function graph
File:R-car stopping distances 1920.svg|Scatter plot
File:Hanger Diagram.png|Hanger diagram.
</gallery>
|}

* Schematics and other types of diagrams, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Train schedule of Sanin Line, Japan, 1949-09-15, part.png|Time–distance diagram
File:Gear pump exploded.png|Exploded view
File:US 2000 census population density map by state.svg|Population density map
File:Pioneer plaque.svg|Pioneer plaque
File:Automotive diagrams 01 En.png|Three-dimensional diagram
</gallery>
|}

=== Table ===
[[File:Table-sample-appearance-default-params-values-01.gif|thumb|300px|An example table rendered in a web browser using HTML.]]
A '''table''' is an arrangement of information or data, typically in rows and columns, or possibly in a more complex structure. Tables are widely used in communication, research, and data analysis. Tables appear in print media, handwritten notes, computer software, architectural ornamentation, traffic signs, and many other places. The precise conventions and terminology for describing tables vary depending on the context. Further, tables differ significantly in variety, structure, flexibility, notation, representation and use. Information or data conveyed in table form is said to be in '''tabular''' format (adjective). In books and technical articles, tables are typically presented apart from the main text in numbered and captioned floating blocks.

==== Basic description ====
A table consists of an ordered arrangement of '''rows''' and '''columns'''. This is a simplified description of the most basic kind of table. Certain considerations follow from this simplified description:

* the term '''row''' has several common synonyms (e.g., record, k-tuple, n-tuple, vector);
* the term '''column''' has several common synonyms (e.g., field, parameter, property, attribute, stanchion);
* a column is usually identified by a name;
* a column name can consist of a word, phrase or a numerical index;
* the intersection of a row and a column is called a cell.

The elements of a table may be grouped, segmented, or arranged in many different ways, and even nested recursively. Additionally, a table may include metadata, annotations, a header, a footer or other ancillary features.

; Simple table
The following illustrates a simple table with three columns and nine rows. The first row is not counted, because it is only used to display the column names. This is called a "header row".

{| class="wikitable" style="text-align:center"
|+Age table
|-
! First name !! Last name !! Age
|-
| Tinu || Elejogun|| 14
|-
| Javier || Zapata || 28
|-
| Lily || McGarrett || 18
|-
| Olatunkbo || Chijiaku || 22
|-
| Adrienne || Anthoula || 22
|-
| Axelia|| Athanasios || 22
|-
| Jon-Kabat || Zinn || 22
|-
| Thabang|| Mosoa||15
|-
| Kgaogelo|| Mosoa||11
|-
|}

; Multi-dimensional table
[[File:Rollup table.png|thumb|An example of a table containing rows with summary information. The summary information consists of subtotals that are combined from previous rows within the same column.]]

The concept of '''dimension''' is also a part of basic terminology. Any "simple" table can be represented as a "multi-dimensional"
table by normalizing the data values into ordered hierarchies. A common example of such a table is a multiplication table.

{| class="wikitable" style="text-align:center"
|+ style="white-space:nowrap" |Multiplication table
|-
!×!! 1 !! 2 !! 3
|-
! 1
| 1 || 2 || 3
|-
! 2
| 2 || 4 || 6
|-
! 3
| 3 || 6 || 9
|}

In multi-dimensional tables, each cell in the body of the table (and the value of that cell) relates to the values at the beginnings of the column (i.e. the header), the row, and other structures in more complex tables. This is an injective relation: each combination of the values of the headers row (row 0, for lack of a better term) and the headers column (column 0 for lack of a better term) is related to a unique cell in
the table:

* Column 1 and row 1 will only correspond to cell (1,1);
* Column 1 and row 2 will only correspond to cell (2,1) etc.

The first column often presents information dimension description by which the rest of the table is navigated. This column is called "stub column". Tables may contain three or multiple dimensions and can be classified by the number of dimensions. Multi-dimensional tables may have super-rows - rows that describe additional dimensions for the rows that are presented below that row and are usually grouped in a tree-like structure. This structure is typically visually presented with an appropriate number of white spaces in front of each stub's label.

In literature tables often present numerical values, cumulative statistics, categorical values, and at times parallel descriptions in form of text. They can condense large amount of information to a limited space and therefore they are popular in scientific literature in many fields of study.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Mathematical_logic Mathematical logic, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Problem_solving Problem solving, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Manipulative_(mathematics_education) Manipulative (mathematics education), Wikipedia] under a CC BY-Sa license
* [https://en.wikipedia.org/wiki/Cuisenaire_rods Cuisenaire rods, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Pictogram Pictogram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Tally_marks Tally marks, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Diagram Diagram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Table_(information) Table (information), Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T19:25:17Z

Khanh: /* Tally marks */

== Formal logical systems ==
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.

=== First-order logic ===
'''First-order logic''' is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.

Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.

Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The '''first incompleteness theorem''' states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The '''second incompleteness theorem''' states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

=== Other classical logics ===
Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''fixed-point logics''' that allow inductive definitions, like one writes for primitive recursive functions.

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

=== Nonclassical and modal logic ===
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

=== Algebraic logic ===
Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.

== Problem-solving strategies ==
Problem-solving strategies are the steps that one would use to find the problems that are in the way to getting to one's own goal. Some refer to this as the "problem-solving cycle".

In this cycle one will acknowledge, recognize the problem, define the problem, develop a strategy to fix the problem, organize the knowledge of the problem cycle, figure out the resources at the user's disposal, monitor one's progress, and evaluate the solution for accuracy. The reason it is called a cycle is that once one is completed with a problem another will usually pop up.

'''Insight''' is the sudden solution to a long-vexing '''problem''', a sudden recognition of a new idea, or a sudden understanding of a complex situation, an ''Aha!'' moment. Solutions found through '''insight''' are often more accurate than those found through step-by-step analysis. To solve more problems at a faster rate, insight is necessary for selecting productive moves at different stages of the problem-solving cycle. This problem-solving strategy pertains specifically to problems referred to as insight problem. Unlike Newell and Simon's formal definition of move problems, there has not been a generally agreed upon definition of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011).

Blanchard-Fields looks at problem solving from one of two facets. The first looking at those problems that only have one solution (like mathematical problems, or fact-based questions) which are grounded in psychometric intelligence. The other is socioemotional in nature and have answers that change constantly (like what's your favorite color or what you should get someone for Christmas).

The following techniques are usually called ''problem-solving strategies''
* Abstraction: solving the problem in a model of the system before applying it to the real system
* Analogy: using a solution that solves an analogous problem
* Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
* Critical thinking
* Divide and conquer: breaking down a large, complex problem into smaller, solvable problems
* Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
* Lateral thinking: approaching solutions indirectly and creatively
* Means-ends analysis: choosing an action at each step to move closer to the goal
* Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new
* Morphological analysis: assessing the output and interactions of an entire system
* Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
* Reduction: transforming the problem into another problem for which solutions exist
* Research: employing existing ideas or adapting existing solutions to similar problems
* Root cause analysis: identifying the cause of a problem
* Trial-and-error: testing possible solutions until the right one is found

== Multiple forms of representation ==
=== Concrete models===
====Base ten blocks====
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

====Pattern blocks====
[[File:Wooden pattern blocks dodecagon.JPG|thumb|One of the ways of making a dodecagon with pattern blocks]]
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.

Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.

Pattern blocks can also serve to provide students with an understanding of fractions. Because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.

Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

====Unifix® Cubes====
[[File:Linking cm cubes 2.JPG|thumb|Interlocking centimeter cubes]]
Unifix® Cubes are interlocking cubes that are just under 2 centimeters on each side. The cubes connect with each other from one side. Once connected, Unifix® Cubes can be turned to form a vertical Unifix® "tower," or horizontally to form a Unifix® "train".

Other interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities.

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence,
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue..
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.

Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume.

====Tiles====
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).

Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.

These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

====Number lines====
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

====Cuisenaire rods====
[[File:Cuisenaire ten.JPG|thumb|Cuisenaire rods used to illustrate the factors of ten]]
'''Cuisenaire rods''' are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods ''réglettes''.

According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."

;Use in mathematics teaching
The rods are used in teaching a variety of mathematical ideas, and with a wide age range of learners. Topics they are used for include:
* Counting, sequences, patterns and algebraic reasoning
* Addition and subtraction (additive reasoning)
* Multiplication and division (multiplicative reasoning)
* Fractions, ratio and proportion
* Modular arithmetic leading to group theory

=== Pictures ===
==== Pictogram ====
[[File:Titanic casualties.svg|thumb|300px|alt=table with boxes instead of numbers, the amounts and sizes of boxes represent amounts of people|A compound pictogram showing the breakdown of the survivors and deaths of the maiden voyage of the RMS Titanic by class and age/gender.]]

Pictograms are charts in which icons represent numbers to make it more interesting and easier to understand. A key is often included to indicate what each icon represents. All icons must be of the same size, but a fraction of an icon can be used to show the respective fraction of that amount.

For example, the following table:
{| class="wikitable"
! Day !! Letters sent
|-
| Monday || 10
|-
| Tuesday || 17
|-
| Wednesday || 29
|-
| Thursday || 41
|-
| Friday || 18
|}
can be graphed as follows:
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+
! Day !! Letters sent
|-
| Monday || [[File:Email Silk.svg|alt=one envelope]]
|-
| Tuesday || [[File:Email Silk.svg|alt=one envelope]] [[File:Image from the Silk icon theme by Mark James half left.svg|alt=and a half]]
|-
| Wednesday || [[File:Email Silk.svg|alt=three envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Thursday || [[File:Email Silk.svg|alt=four envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Friday || [[File:Email Silk.svg|alt=two envelopes]] [[File:Email Silk.svg|alt=]]
|}
Key: [[File:Email Silk.svg|alt=one envelope]] = 10 letters

As the values are rounded to the nearest 5 letters, the second icon on Tuesday is the left half of the original.

==== Tally marks ====
[[File:Strike symbol (49778996432).jpg |thumb|Tally marks on a chalkboard]]
'''Tally marks''', also called '''hash marks''', are a unary numeral system. They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded.

However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were also historically used for this purpose.

;Clustering
Tally marks are typically clustered in groups of five for legibility. The cluster size 5 has the advantages of (a) easy conversion into decimal for higher arithmetic operations and (b) avoiding error, as humans can far more easily correctly identify a cluster of 5 than one of 10.

<gallery>
File:Tally marks.svg|Tally marks high in most of Europe, Zimbabwe and Southern Africa, Australia, New Zealand and North America. In some variants, the diagonal/horizontal slash is used on its own when five or more units are added at once.
File:Tally marks 3.svg|Cultures using Chinese characters tally by forming the character 正, which consists of five strokes.
File:Tally marks 2.svg|Tally marks used in France, their former colonies, Argentina and Brazil. 1 to 5 and so on. These are most commonly used for registering scores in card games, like Truco
File:Dot and line tally marks.jpg|In the dot and line (or dot-dash) tally, dots represent counts from 1 to 4, lines 5 to 8, and diagonal lines 9 and 10. This method is commonly used in forestry and related fields.
</gallery>

=== Diagrams ===
A '''diagram''' is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram.

====Gallery of diagram types====

There are at least the following types of diagrams:

* Logical or conceptual diagrams, which take a collection of items and relationships between them, and express them by giving each item a 2D position, while the relationships are expressed as connections between the items or overlaps between the items, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Tree Example.png|Tree diagram
File:Neural network.svg|Network diagram
File:LampFlowchart.svg|Flowchart
File:Set intersection.svg|Venn diagram
File:Alphagraphen.png|Existential graph
</gallery>
|}

* Quantitative diagrams, which display a relationship between two variables that take either discrete or a continuous range of values; for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Histogram example.svg|Histogram
File:Graphtestone.svg|Bar graph
File:Zusammensetzung Shampoo.svg|Pie chart
File:Hyperbolic Cosine.svg|Function graph
File:R-car stopping distances 1920.svg|Scatter plot
File:Hanger Diagram.png|Hanger diagram.
</gallery>
|}

* Schematics and other types of diagrams, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Train schedule of Sanin Line, Japan, 1949-09-15, part.png|Time–distance diagram
File:Gear pump exploded.png|Exploded view
File:US 2000 census population density map by state.svg|Population density map
File:Pioneer plaque.svg|Pioneer plaque
File:Automotive diagrams 01 En.png|Three-dimensional diagram
</gallery>
|}

=== Table ===
[[File:Table-sample-appearance-default-params-values-01.gif|thumb|300px|An example table rendered in a web browser using HTML.]]
A '''table''' is an arrangement of information or data, typically in rows and columns, or possibly in a more complex structure. Tables are widely used in communication, research, and data analysis. Tables appear in print media, handwritten notes, computer software, architectural ornamentation, traffic signs, and many other places. The precise conventions and terminology for describing tables vary depending on the context. Further, tables differ significantly in variety, structure, flexibility, notation, representation and use. Information or data conveyed in table form is said to be in '''tabular''' format (adjective). In books and technical articles, tables are typically presented apart from the main text in numbered and captioned floating blocks.

==== Basic description ====
A table consists of an ordered arrangement of '''rows''' and '''columns'''. This is a simplified description of the most basic kind of table. Certain considerations follow from this simplified description:

* the term '''row''' has several common synonyms (e.g., record, k-tuple, n-tuple, vector);
* the term '''column''' has several common synonyms (e.g., field, parameter, property, attribute, stanchion);
* a column is usually identified by a name;
* a column name can consist of a word, phrase or a numerical index;
* the intersection of a row and a column is called a cell.

The elements of a table may be grouped, segmented, or arranged in many different ways, and even nested recursively. Additionally, a table may include metadata, annotations, a header, a footer or other ancillary features.

; Simple table
The following illustrates a simple table with three columns and nine rows. The first row is not counted, because it is only used to display the column names. This is called a "header row".

{| class="wikitable" style="text-align:center"
|+Age table
|-
! First name !! Last name !! Age
|-
| Tinu || Elejogun|| 14
|-
| Javier || Zapata || 28
|-
| Lily || McGarrett || 18
|-
| Olatunkbo || Chijiaku || 22
|-
| Adrienne || Anthoula || 22
|-
| Axelia|| Athanasios || 22
|-
| Jon-Kabat || Zinn || 22
|-
| Thabang|| Mosoa||15
|-
| Kgaogelo|| Mosoa||11
|-
|}

; Multi-dimensional table
[[File:Rollup table.png|thumb|An example of a table containing rows with summary information. The summary information consists of subtotals that are combined from previous rows within the same column.]]

The concept of '''dimension''' is also a part of basic terminology. Any "simple" table can be represented as a "multi-dimensional"
table by normalizing the data values into ordered hierarchies. A common example of such a table is a multiplication table.

{| class="wikitable" style="text-align:center"
|+ style="white-space:nowrap" |Multiplication table
|-
!×!! 1 !! 2 !! 3
|-
! 1
| 1 || 2 || 3
|-
! 2
| 2 || 4 || 6
|-
! 3
| 3 || 6 || 9
|}

In multi-dimensional tables, each cell in the body of the table (and the value of that cell) relates to the values at the beginnings of the column (i.e. the header), the row, and other structures in more complex tables. This is an injective relation: each combination of the values of the headers row (row 0, for lack of a better term) and the headers column (column 0 for lack of a better term) is related to a unique cell in
the table:

* Column 1 and row 1 will only correspond to cell (1,1);
* Column 1 and row 2 will only correspond to cell (2,1) etc.

The first column often presents information dimension description by which the rest of the table is navigated. This column is called "stub column". Tables may contain three or multiple dimensions and can be classified by the number of dimensions. Multi-dimensional tables may have super-rows - rows that describe additional dimensions for the rows that are presented below that row and are usually grouped in a tree-like structure. This structure is typically visually presented with an appropriate number of white spaces in front of each stub's label.

In literature tables often present numerical values, cumulative statistics, categorical values, and at times parallel descriptions in form of text. They can condense large amount of information to a limited space and therefore they are popular in scientific literature in many fields of study.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Mathematical_logic Mathematical logic, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Problem_solving Problem solving, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Manipulative_(mathematics_education) Manipulative (mathematics education), Wikipedia] under a CC BY-Sa license
* [https://en.wikipedia.org/wiki/Cuisenaire_rods Cuisenaire rods, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Pictogram Pictogram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Diagram Diagram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Table_(information) Table (information), Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T19:23:37Z

Khanh: /* Pictures */

== Formal logical systems ==
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.

=== First-order logic ===
'''First-order logic''' is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.

Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.

Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The '''first incompleteness theorem''' states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The '''second incompleteness theorem''' states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

=== Other classical logics ===
Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''fixed-point logics''' that allow inductive definitions, like one writes for primitive recursive functions.

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

=== Nonclassical and modal logic ===
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

=== Algebraic logic ===
Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.

== Problem-solving strategies ==
Problem-solving strategies are the steps that one would use to find the problems that are in the way to getting to one's own goal. Some refer to this as the "problem-solving cycle".

In this cycle one will acknowledge, recognize the problem, define the problem, develop a strategy to fix the problem, organize the knowledge of the problem cycle, figure out the resources at the user's disposal, monitor one's progress, and evaluate the solution for accuracy. The reason it is called a cycle is that once one is completed with a problem another will usually pop up.

'''Insight''' is the sudden solution to a long-vexing '''problem''', a sudden recognition of a new idea, or a sudden understanding of a complex situation, an ''Aha!'' moment. Solutions found through '''insight''' are often more accurate than those found through step-by-step analysis. To solve more problems at a faster rate, insight is necessary for selecting productive moves at different stages of the problem-solving cycle. This problem-solving strategy pertains specifically to problems referred to as insight problem. Unlike Newell and Simon's formal definition of move problems, there has not been a generally agreed upon definition of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011).

Blanchard-Fields looks at problem solving from one of two facets. The first looking at those problems that only have one solution (like mathematical problems, or fact-based questions) which are grounded in psychometric intelligence. The other is socioemotional in nature and have answers that change constantly (like what's your favorite color or what you should get someone for Christmas).

The following techniques are usually called ''problem-solving strategies''
* Abstraction: solving the problem in a model of the system before applying it to the real system
* Analogy: using a solution that solves an analogous problem
* Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
* Critical thinking
* Divide and conquer: breaking down a large, complex problem into smaller, solvable problems
* Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
* Lateral thinking: approaching solutions indirectly and creatively
* Means-ends analysis: choosing an action at each step to move closer to the goal
* Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new
* Morphological analysis: assessing the output and interactions of an entire system
* Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
* Reduction: transforming the problem into another problem for which solutions exist
* Research: employing existing ideas or adapting existing solutions to similar problems
* Root cause analysis: identifying the cause of a problem
* Trial-and-error: testing possible solutions until the right one is found

== Multiple forms of representation ==
=== Concrete models===
====Base ten blocks====
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

====Pattern blocks====
[[File:Wooden pattern blocks dodecagon.JPG|thumb|One of the ways of making a dodecagon with pattern blocks]]
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.

Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.

Pattern blocks can also serve to provide students with an understanding of fractions. Because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.

Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

====Unifix® Cubes====
[[File:Linking cm cubes 2.JPG|thumb|Interlocking centimeter cubes]]
Unifix® Cubes are interlocking cubes that are just under 2 centimeters on each side. The cubes connect with each other from one side. Once connected, Unifix® Cubes can be turned to form a vertical Unifix® "tower," or horizontally to form a Unifix® "train".

Other interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities.

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence,
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue..
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.

Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume.

====Tiles====
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).

Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.

These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

====Number lines====
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

====Cuisenaire rods====
[[File:Cuisenaire ten.JPG|thumb|Cuisenaire rods used to illustrate the factors of ten]]
'''Cuisenaire rods''' are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods ''réglettes''.

According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."

;Use in mathematics teaching
The rods are used in teaching a variety of mathematical ideas, and with a wide age range of learners. Topics they are used for include:
* Counting, sequences, patterns and algebraic reasoning
* Addition and subtraction (additive reasoning)
* Multiplication and division (multiplicative reasoning)
* Fractions, ratio and proportion
* Modular arithmetic leading to group theory

=== Pictures ===
==== Pictogram ====
[[File:Titanic casualties.svg|thumb|300px|alt=table with boxes instead of numbers, the amounts and sizes of boxes represent amounts of people|A compound pictogram showing the breakdown of the survivors and deaths of the maiden voyage of the RMS Titanic by class and age/gender.]]

Pictograms are charts in which icons represent numbers to make it more interesting and easier to understand. A key is often included to indicate what each icon represents. All icons must be of the same size, but a fraction of an icon can be used to show the respective fraction of that amount.

For example, the following table:
{| class="wikitable"
! Day !! Letters sent
|-
| Monday || 10
|-
| Tuesday || 17
|-
| Wednesday || 29
|-
| Thursday || 41
|-
| Friday || 18
|}
can be graphed as follows:
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+
! Day !! Letters sent
|-
| Monday || [[File:Email Silk.svg|alt=one envelope]]
|-
| Tuesday || [[File:Email Silk.svg|alt=one envelope]] [[File:Image from the Silk icon theme by Mark James half left.svg|alt=and a half]]
|-
| Wednesday || [[File:Email Silk.svg|alt=three envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Thursday || [[File:Email Silk.svg|alt=four envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Friday || [[File:Email Silk.svg|alt=two envelopes]] [[File:Email Silk.svg|alt=]]
|}
Key: [[File:Email Silk.svg|alt=one envelope]] = 10 letters

As the values are rounded to the nearest 5 letters, the second icon on Tuesday is the left half of the original.

==== Tally marks ====
[[File:Strike symbol (49778996432).jpg |thumb|Tally marks on a chalkboard]]
[[File:Hanakapiai Beach Warning Sign Only.jpg|upright|thumb|Counting using tally marks at Hanakapiai Beach. The number shown is 82.]]

'''Tally marks''', also called '''hash marks''', are a unary numeral system. They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded.

However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were also historically used for this purpose.

;Clustering
[[File:Unary8.svg|thumb|upright|150px|Various ways to cluster the number 8. The first or fifth mark in each group may be written at an angle to the others for easier distinction. In the fourth example, the fifth stroke "closes out" a group of five, forming a "herringbone". In the fifth row the fifth mark crosses diagonally, forming a "five-bar gate".]]

Tally marks are typically clustered in groups of five for legibility. The cluster size 5 has the advantages of (a) easy conversion into decimal for higher arithmetic operations and (b) avoiding error, as humans can far more easily correctly identify a cluster of 5 than one of 10.

<gallery>
File:Tally marks.svg|Tally marks high in most of Europe, Zimbabwe and Southern Africa, Australia, New Zealand and North America. In some variants, the diagonal/horizontal slash is used on its own when five or more units are added at once.
File:Tally marks 3.svg|Cultures using Chinese characters tally by forming the character 正, which consists of five strokes.
File:Tally marks 2.svg|Tally marks used in France, their former colonies, Argentina and Brazil. 1 to 5 and so on. These are most commonly used for registering scores in card games, like Truco
File:Dot and line tally marks.jpg|In the dot and line (or dot-dash) tally, dots represent counts from 1 to 4, lines 5 to 8, and diagonal lines 9 and 10. This method is commonly used in forestry and related fields.
</gallery>

=== Diagrams ===
A '''diagram''' is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram.

====Gallery of diagram types====

There are at least the following types of diagrams:

* Logical or conceptual diagrams, which take a collection of items and relationships between them, and express them by giving each item a 2D position, while the relationships are expressed as connections between the items or overlaps between the items, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Tree Example.png|Tree diagram
File:Neural network.svg|Network diagram
File:LampFlowchart.svg|Flowchart
File:Set intersection.svg|Venn diagram
File:Alphagraphen.png|Existential graph
</gallery>
|}

* Quantitative diagrams, which display a relationship between two variables that take either discrete or a continuous range of values; for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Histogram example.svg|Histogram
File:Graphtestone.svg|Bar graph
File:Zusammensetzung Shampoo.svg|Pie chart
File:Hyperbolic Cosine.svg|Function graph
File:R-car stopping distances 1920.svg|Scatter plot
File:Hanger Diagram.png|Hanger diagram.
</gallery>
|}

* Schematics and other types of diagrams, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Train schedule of Sanin Line, Japan, 1949-09-15, part.png|Time–distance diagram
File:Gear pump exploded.png|Exploded view
File:US 2000 census population density map by state.svg|Population density map
File:Pioneer plaque.svg|Pioneer plaque
File:Automotive diagrams 01 En.png|Three-dimensional diagram
</gallery>
|}

=== Table ===
[[File:Table-sample-appearance-default-params-values-01.gif|thumb|300px|An example table rendered in a web browser using HTML.]]
A '''table''' is an arrangement of information or data, typically in rows and columns, or possibly in a more complex structure. Tables are widely used in communication, research, and data analysis. Tables appear in print media, handwritten notes, computer software, architectural ornamentation, traffic signs, and many other places. The precise conventions and terminology for describing tables vary depending on the context. Further, tables differ significantly in variety, structure, flexibility, notation, representation and use. Information or data conveyed in table form is said to be in '''tabular''' format (adjective). In books and technical articles, tables are typically presented apart from the main text in numbered and captioned floating blocks.

==== Basic description ====
A table consists of an ordered arrangement of '''rows''' and '''columns'''. This is a simplified description of the most basic kind of table. Certain considerations follow from this simplified description:

* the term '''row''' has several common synonyms (e.g., record, k-tuple, n-tuple, vector);
* the term '''column''' has several common synonyms (e.g., field, parameter, property, attribute, stanchion);
* a column is usually identified by a name;
* a column name can consist of a word, phrase or a numerical index;
* the intersection of a row and a column is called a cell.

The elements of a table may be grouped, segmented, or arranged in many different ways, and even nested recursively. Additionally, a table may include metadata, annotations, a header, a footer or other ancillary features.

; Simple table
The following illustrates a simple table with three columns and nine rows. The first row is not counted, because it is only used to display the column names. This is called a "header row".

{| class="wikitable" style="text-align:center"
|+Age table
|-
! First name !! Last name !! Age
|-
| Tinu || Elejogun|| 14
|-
| Javier || Zapata || 28
|-
| Lily || McGarrett || 18
|-
| Olatunkbo || Chijiaku || 22
|-
| Adrienne || Anthoula || 22
|-
| Axelia|| Athanasios || 22
|-
| Jon-Kabat || Zinn || 22
|-
| Thabang|| Mosoa||15
|-
| Kgaogelo|| Mosoa||11
|-
|}

; Multi-dimensional table
[[File:Rollup table.png|thumb|An example of a table containing rows with summary information. The summary information consists of subtotals that are combined from previous rows within the same column.]]

The concept of '''dimension''' is also a part of basic terminology. Any "simple" table can be represented as a "multi-dimensional"
table by normalizing the data values into ordered hierarchies. A common example of such a table is a multiplication table.

{| class="wikitable" style="text-align:center"
|+ style="white-space:nowrap" |Multiplication table
|-
!×!! 1 !! 2 !! 3
|-
! 1
| 1 || 2 || 3
|-
! 2
| 2 || 4 || 6
|-
! 3
| 3 || 6 || 9
|}

In multi-dimensional tables, each cell in the body of the table (and the value of that cell) relates to the values at the beginnings of the column (i.e. the header), the row, and other structures in more complex tables. This is an injective relation: each combination of the values of the headers row (row 0, for lack of a better term) and the headers column (column 0 for lack of a better term) is related to a unique cell in
the table:

* Column 1 and row 1 will only correspond to cell (1,1);
* Column 1 and row 2 will only correspond to cell (2,1) etc.

The first column often presents information dimension description by which the rest of the table is navigated. This column is called "stub column". Tables may contain three or multiple dimensions and can be classified by the number of dimensions. Multi-dimensional tables may have super-rows - rows that describe additional dimensions for the rows that are presented below that row and are usually grouped in a tree-like structure. This structure is typically visually presented with an appropriate number of white spaces in front of each stub's label.

In literature tables often present numerical values, cumulative statistics, categorical values, and at times parallel descriptions in form of text. They can condense large amount of information to a limited space and therefore they are popular in scientific literature in many fields of study.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Mathematical_logic Mathematical logic, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Problem_solving Problem solving, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Manipulative_(mathematics_education) Manipulative (mathematics education), Wikipedia] under a CC BY-Sa license
* [https://en.wikipedia.org/wiki/Cuisenaire_rods Cuisenaire rods, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Pictogram Pictogram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Diagram Diagram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Table_(information) Table (information), Wikipedia] under a CC BY-SA license

Number Systems, Base 10, 5 and 2

2022-02-05T19:17:20Z

Khanh: Created page with "Numbers written in different numeral systems. A '''numeral system''' (or '''system of numeration''') is a writin..."

[[File:Numeral Systems of the World.svg|264px|thumb|right|Numbers written in different numeral systems.]]

A '''numeral system''' (or '''system of numeration''') is a [[writing system]] for expressing numbers; that is, a [[mathematical notation]] for representing [[number]]s of a given set, using [[Numerical digit|digits]] or other symbols in a consistent manner.

The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number ''eleven'' in the [[decimal numeral system]] (used in common life), the number ''three'' in the [[binary numeral system]] (used in [[computer]]s), and the number ''two'' in the [[unary numeral system]] (e.g. used in [[Tally marks|tallying]] scores).

The number the numeral represents is called its value. Not all number systems can represent all numbers that are considered in the modern days; for example, Roman numerals have no zero.

Ideally, a numeral system will:
*Represent a useful set of numbers (e.g. all [[integer]]s, or [[rational number]]s)
*Give every number represented a unique representation (or at least a standard representation)
*Reflect the [[algebra|algebraic]] and [[arithmetic]] structure of the numbers.

For example, the usual [[decimal representation]] gives every nonzero [[natural number]] a unique representation as a [[finite set|finite]] [[sequence]] of [[numerical digit|digits]], beginning with a non-zero digit.

Numeral systems are sometimes called ''[[number system]]s'', but that name is ambiguous, as it could refer to different systems of numbers, such as the system of [[real number]]s, the system of [[complex number]]s, the system of [[p-adic number|''p''-adic numbers]], etc. Such systems are, however, not the topic of this article.

==Main numeral systems==
The most commonly used system of numerals is [[decimal]]. [[Indian mathematicians]] are credited with developing the integer version, the [[Hindu–Arabic numeral system]]. [[Aryabhata]] of [[Patna|Kusumapura]] developed the [[place-value notation]] in the 5th century and a century later [[Brahmagupta]] introduced the symbol for [[zero]]. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. [[Middle-East|Middle-Eastern]] mathematicians extended the system to include negative powers of 10 ([[fractions]]), as recorded in a treatise by [[Syrian]] mathematician [[Abu'l-Hasan al-Uqlidisi]] in 952–953, and the [[decimal point]] notation was introduced{{when|date=February 2021}} by [[Sind ibn Ali]], who also wrote the earliest treatise on Arabic numerals. The Hindu-Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called [[Arabic numerals]], as they learned them from the Arabs.

The simplest numeral system is the [[unary numeral system]], in which every [[natural number]] is represented by a corresponding number of symbols. If the symbol {{mono|/}} is chosen, for example, then the number seven would be represented by {{mono|///////}}. [[Tally marks]] represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in [[theoretical computer science]]. [[Elias gamma coding]], which is commonly used in [[data compression]], expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.

The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as {{mono|+++ ////}} and the number 123 as {{mono|+ − − ///}} without any need for zero. This is called [[sign-value notation]]. The ancient [[Egyptian numeral system]] was of this type, and the [[Roman numeral system]] was a modification of this idea.

More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of the alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, one could then write C+ D/ for the number 304. This system is used when writing [[Chinese numerals]] and other East Asian numerals based on Chinese. The number system of the [[English language]] is of this type ("three hundred [and] four"), as are those of other spoken [[language]]s, regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for instance 79 in French is ''soixante dix-neuf'' ({{nowrap|60 + 10 + 9}}) and in Welsh is ''pedwar ar bymtheg a thrigain'' ({{nowrap|4 + (5 + 10) + (3 × 20)}}) or (somewhat archaic) ''pedwar ugain namyn un'' ({{nowrap|4 × 20 − 1}}). In English, one could say "four score less one", as in the famous [[Gettysburg Address]] representing "87 years ago" as "four score and seven years ago".

More elegant is a ''[[positional notation|positional system]]'', also known as place-value notation. Again working in base 10, ten different digits 0, ..., 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in {{nowrap|304 {{=}} 3×100 + 0×10 + 4×1}} or more precisely {{nowrap|3×102 + 0×101 + 4×100}}. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base 10).<ref>{{Cite book|last=Chowdhury|first=Arnab|url=https://books.google.com/books?id=WXn-mT3K6dgC&q=Arithmetic+is+much+easier+in+positional+systems+than+in+the+earlier+additive+ones;+furthermore,+additive+systems+need+a+large+number+of+different+symbols+for+the+different+powers+of+10;+a+positional+system+needs+only+ten+different+symbols+(assuming+that+it+uses+base+10).&pg=PA2|title=Design of an Efficient Multiplier using DBNS|publisher=GIAP Journals|isbn=978-93-83006-18-2|language=en}}</ref>

The positional decimal system is presently universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers.

In [[computers]], the main numeral systems are based on the positional system in base 2 ([[binary numeral system]]), with two [[binary digit]]s, 0 and 1. Positional systems obtained by grouping binary digits by three ([[octal numeral system]]) or four ([[hexadecimal numeral system]]) are commonly used. For very large integers, bases 232 or 264 (grouping binary digits by 32 or 64, the length of the [[machine word]]) are used, as, for example, in [[GNU Multiple Precision Arithmetic Library|GMP]].

In certain biological systems, the [[unary coding]] system is employed. Unary numerals used in the [[neural circuit]]s responsible for [[birdsong]] production.<ref> Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New Encyclopedia of Neuroscience.</ref> The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC ([[high vocal center]]). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.

The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the [[Arithmetic sequence|arithmetic]] numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the [[Geometric sequence|geometric]] numerals (1, 10, 100, 1000, 10000 ...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the [[Greek numerals|Ionic system]]), and a positional system does not need geometric numerals because they are made by position. However, the spoken language uses ''both'' arithmetic and geometric numerals.

In certain areas of computer science, a modified base ''k'' positional system is used, called [[bijective numeration]], with digits 1, 2, ..., ''k'' ({{nowrap|''k'' ≥ 1}}), and zero being represented by an empty string. This establishes a [[bijection]] between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-''k'' numeration is also called ''k''-adic notation, not to be confused with [[p-adic number|''p''-adic numbers]]. Bijective base 1 is the same as unary.

==Positional systems in detail==
In a positional base ''b'' numeral system (with ''b'' a [[natural number]] greater than 1 known as the [[radix]]), ''b'' basic symbols (or digits) corresponding to the first ''b'' natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by ''b''.

For example, in the [[decimal]] system (base 10), the numeral 4327 means {{math|('''4'''×103) + ('''3'''×102) + ('''2'''×101) + ('''7'''×100)}}, noting that {{math|100 {{=}} 1}}.

In general, if ''b'' is the base, one writes a number in the numeral system of base ''b'' by expressing it in the form {{math|''a''''n''''b''''n'' + ''a''''n'' − 1''b''''n'' − 1 + ''a''''n'' − 2''b''''n'' − 2 + ... + ''a''0''b''0}} and writing the enumerated digits {{math|''a''''n''''a''''n'' − 1''a''''n'' − 2 ... ''a''0}} in descending order. The digits are natural numbers between 0 and {{math|''b'' − 1}}, inclusive.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: numberbase. Unless specified by context, numbers without subscript are considered to be decimal.

By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base 2 numeral 10.11 denotes {{math|1×21 + 0×20 + 1×2−1 + 1×2−2 {{=}} 2.75}}.

In general, numbers in the base ''b'' system are of the form:

:<math>
(a_na_{n-1}\cdots a_1a_0.c_1 c_2 c_3\cdots)_b =
\sum_{k=0}^n a_kb^k + \sum_{k=1}^\infty c_kb^{-k}.
</math>

The numbers ''b''''k'' and ''b''−''k'' are the [[weight function|weights]] of the corresponding digits. The position ''k'' is the [[logarithm]] of the corresponding weight ''w'', that is <math>k = \log_{b} w = \log_{b} b^k</math>. The highest used position is close to the [[order of magnitude]] of the number.

The number of [[tally marks]] required in the [[unary numeral system]] for ''describing the weight'' would have been '''w'''. In the positional system, the number of digits required to describe it is only <math>k + 1 = \log_{b} w + 1</math>, for ''k'' ≥ 0. For example, to describe the weight 1000 then four digits are needed because <math>\log_{10} 1000 + 1 = 3 + 1</math>. The number of digits required to ''describe the position'' is <math>\log_b k + 1 = \log_b \log_b w + 1</math> (in positions 1, 10, 100,... only for simplicity in the decimal example).

:<math>\begin{array}{l|rrrrrrr}
\text{Position}
& 3
& 2
& 1
& 0
& -1
& -2
& \cdots
\\
\hline
\text{Weight}
& b^3 & b^2 & b^1 & b^0 & b^{-1} & b^{-2} & \cdots
\\
\text{Digit}
& a_3 & a_2 & a_1 & a_0 & c_1 & c_2 & \cdots
\\
\hline
\text{Decimal example weight}
& 1000 & 100 & 10 & 1 & 0.1 & 0.01 & \cdots
\\
\text{Decimal example digit}
& 4 & 3 & 2 & 7 & 0 & 0 & \cdots
\end{array}
</math>

A number has a terminating or repeating expansion [[if and only if]] it is [[rational number|rational]]; this does not depend on the base. A number that terminates in one base may repeat in another (thus {{math|0.310 {{=}} 0.0100110011001...2}}). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, {{math|[[pi|π]] {{=}} 3.1415926...10}} can be written as the aperiodic 11.001001000011111...2.

Putting [[overline|overscores]], {{overline|''n''}}, or dots, ''ṅ'', above the common digits is a convention used to represent repeating rational expansions. Thus:
:14/11 = 1.272727272727... = 1.{{overline|27}}   or   321.3217878787878... = 321.321{{Overline|78}}.

If ''b'' = ''p'' is a prime number, one can define base-''p'' numerals whose expansion to the left never stops; these are called the ''p''-adic numbers.

==Generalized variable-length integers==
More general is using a mixed radix notation (here written little-endian) like <math>a_0 a_1 a_2</math> for <math>a_0 + a_1 b_1 + a_2 b_1 b_2</math>, etc.

This is used in Punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a–z and 0–9, representing 0–25 and 26–35 respectively. There are also so-called threshold values (<math>t_0, t_1, ...</math>) which are fixed for every position in the number. A digit <math>a_i</math> (in a given position in the number) that is lower than its corresponding threshold value <math>t_i</math> means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next number.

For example, if the threshold value for the first digit is ''b'' (i.e. 1) then ''a'' (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit, first-digit range is only b–9 (i.e. 1–35), therefore the weight ''b''1 is 35 instead of 36. More generally, if ''tn'' is the threshold for the ''n''-th digit, it is easy to show that <math>b_{n+1}=36-t_n</math>.
Suppose the threshold values for the second and third digits are ''c'' (i.e. 2), then the second-digit range is a–b (i.e. 0–1) with the second digit being most significant, while the range is c–9 (i.e. 2–35) in the presence of a third digit. Generally, for any ''n'', the weight of the (''n''+1)-th digit is the weight of the previous one times (36 − threshold of the ''n''-th digit). So the weight of the second symbol is <math>36 - t_0 = 35</math>. And the weight of the third symbol is <math>35 * (36 - t_1) = 35*34 = 1190</math>.

So we have the following sequence of the numbers with at most 3 digits:

''a'' (0), ''ba'' (1), ''ca'' (2), ..., ''9a'' (35), ''bb'' (36), ''cb'' (37), ..., ''9b'' (70), ''bca'' (71), ..., ''99a'' (1260), ''bcb'' (1261), ..., ''99b'' (2450).

Unlike a regular n-based numeral system, there are numbers like ''9b'' where ''9'' and ''b'' each represent 35; yet the representation is unique because ''ac'' and ''aca'' are not allowed – the first ''a'' would terminate each of the se numbers.

The flexibility in choosing threshold values allows optimization for number of digits depending on the frequency of occurrence of numbers of various sizes.

The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are non-zero.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Numeral_system Numeral system, Wikipedia] under a CC BY-SA license

Base 10, Base 2 & Base 5

2022-02-05T19:13:12Z

Khanh: /* Fractions */

In a positional numeral system, the '''radix''' or '''base''' is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9.

In any standard positional numeral system, a number is conventionally written as (''x'')''y'' with ''x'' as the string of digits and ''y'' as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)10 is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.

= In numeral systems =
In the system with radix 13, for example, a string of digits such as 398 denotes the (decimal) number 3 × 132 + 9 × 131 + 8 × 130 = 632.

More generally, in a system with radix ''b'' (''b'' > 1), a string of digits ''d''1 … ''dn'' denotes the number ''d''1''b''''n''−1 + ''d''2''b''''n''−2 + … + ''dnb''0, where 0 ≤ ''di'' < ''b''. In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radix ''b'' would have a ones' place, then a ''b''1s' place, a ''b''2s' place, etc.

Commonly used numeral systems include:
{| class="wikitable sortable"
! Base/radix
! Name
! Description
|-
| 2
| Binary numeral system
| Used internally by nearly all computers, is base 2. The two digits are "0" and "1", expressed from switches displaying OFF and ON, respectively. Used in most electric counters.
|-
| 8
| Octal system
| Used occasionally in computing. The eight digits are "0"–"7" and represent 3 bits (23).
|-
| 10
| Decimal system
| Used by humans in the vast majority of cultures. Its ten digits are "0"–"9". Used in most mechanical counters.
|-
| 12
| Duodecimal (dozenal) system
| Sometimes advocated due to divisibility by 2, 3, 4, and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
|-
| 16
| Hexadecimal system
| Often used in computing as a more compact representation of binary (1 hex digit per 4 bits). The sixteen digits are "0"–"9" followed by "A"–"F" or "a"–"f".
|-
| 20
| Vigesimal system
| Traditional numeral system in several cultures, still used by some for counting. Historically also known as the ''score system'' in English, now most famous in the phrase "four score and seven years ago" in the Gettysburg Address.
|-
| 60
| Sexagesimal system
| Originated in ancient Sumer and passed to the Babylonians. Used today as the basis of modern circular coordinate system (degrees, minutes, and seconds) and time measuring (minutes, and seconds) by analogy to the rotation of the Earth.
|}

= Base 2 =
A '''binary number''' is a number expressed in the '''base-2 numeral system''' or '''binary numeral system''', a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).

The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language.

==Representation==
Any number can be represented by a sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. Any of the following rows of symbols can be interpreted as the binary numeric value of 667:

{| style="text-align:center;"
| 1 || 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1 || 1
|-
| <nowiki>|</nowiki> || ― || <nowiki>|</nowiki> || ― || ― || <nowiki>|</nowiki> || <nowiki>|</nowiki> || ― || <nowiki>|</nowiki> || <nowiki>|</nowiki>
|-
| ☒ || ☐ || ☒ || ☐ || ☐ || ☒ || ☒ || ☐ || ☒ || ☒
|-
| y || n || y || n || n || y || y || n || y || y
|}

[[Image:Binary clock.svg|250px|thumbnail|right|A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.]]

The numeric value represented in each case is dependent upon the value assigned to each symbol. In the earlier days of computing, switches, punched holes and punched paper tapes were used to represent binary values. In a modern computer, the numeric values may be represented by two different voltages; on a magnetic disk, magnetic polarities may be used. A "positive", "yes", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.

In keeping with customary representation of numerals using Arabic numerals, binary numbers are commonly written using the symbols '''0''' and '''1'''. When written, binary numerals are often subscripted, prefixed or suffixed in order to indicate their base, or radix. The following notations are equivalent:
* 100101 binary (explicit statement of format)
* 100101b (a suffix indicating binary format; also known as Intel convention)
* 100101B (a suffix indicating binary format)
* bin 100101 (a prefix indicating binary format)
* 1001012 (a subscript indicating base-2 (binary) notation)
* %100101 (a prefix indicating binary format; also known as Motorola convention)
* 0b100101 (a prefix indicating binary format, common in programming languages)
* 6b100101 (a prefix indicating number of bits in binary format, common in programming languages)
* #b100101 (a prefix indicating binary format, common in Lisp programming languages)

When spoken, binary numerals are usually read digit-by-digit, in order to distinguish them from decimal numerals. For example, the binary numeral 100 is pronounced ''one zero zero'', rather than ''one hundred'', to make its binary nature explicit, and for purposes of correctness. Since the binary numeral 100 represents the value four, it would be confusing to refer to the numeral as ''one hundred'' (a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct ''value''), but this does not make its binary nature explicit.

==Counting in binary==
Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar decimal counting system as a frame of reference.

===Decimal counting===
Decimal counting uses the ten symbols ''0'' through ''9''. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the ''first digit''. When the available symbols for this position are exhausted, the least significant digit is reset to ''0'', and the next digit of higher significance (one position to the left) is incremented (''overflow''), and incremental substitution of the low-order digit resumes. This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows:

:000, 001, 002, ... 007, 008, 009, (rightmost digit is reset to zero, and the digit to its left is incremented)
:0'''1'''0, 011, 012, ...
:   ...
:090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented)
:'''1'''00, 101, 102, ...

===Binary counting===
[[File:Binary counter.gif|thumb|This counter shows how to count in binary from numbers zero through thirty-one.]]
[[File:Binary_guess_number_trick_SMIL.svg|thumb|upright|link={{filepath:binary_guess_number_trick_SMIL.svg}}|A party trick to guess a number from which cards it is printed on uses the bits of the binary representation of the number. In the SVG file, click a card to toggle it]]
Binary counting follows the same procedure, except that only the two symbols ''0'' and ''1'' are available. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:

:0000,
:000'''1''', (rightmost digit starts over, and next digit is incremented)
:00'''1'''0, 0011, (rightmost two digits start over, and next digit is incremented)
:0'''1'''00, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented)
:'''1'''000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...

In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" digit. For example, the binary number 100101 is converted to decimal form as follows:

:1001012 = [ ( '''1''' ) × 25 ] + [ ( '''0''' ) × 24 ] + [ ( '''0''' ) × 23 ] + [ ( '''1''' ) × 22 ] + [ ( '''0''' ) × 21 ] + [ ( '''1''' ) × 20 ]

:1001012 = [ '''1''' × 32 ] + [ '''0''' × 16 ] + [ '''0''' × 8 ] + [ '''1''' × 4 ] + [ '''0''' × 2 ] + [ '''1''' × 1 ]

:'''1001012 = 3710'''

==Fractions==

Fractions in binary arithmetic terminate only if 2 is the only prime factor in the denominator. As a result, 1/10 does not have a finite binary representation ('''10''' has prime factors '''2''' and '''5'''). This causes 10 × 0.1 not to precisely equal 1 in floating-point arithmetic. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × '''2−1''' + 1 × '''2−2''' + 0 × '''2−3''' + 1 × '''2−4''' + ... = 0.3125 + ... An exact value cannot be found with a sum of a finite number of inverse powers of two, the zeros and ones in the binary representation of 1/3 alternate forever.

{| class="wikitable"
|-
! Fraction
! Decimal
! Binary
! Fractional approximation
|-
| 1/1
| 1 or 0.999...
| 1 or 0.111...
| 1/2 + 1/4 + 1/8...
|-
| 1/2
| 0.5 or 0.4999...
| 0.1 or 0.0111...
| 1/4 + 1/8 + 1/16 . . .
|-
| 1/3
| 0.333...
| 0.010101...
| 1/4 + 1/16 + 1/64 . . .
|-
| 1/4
| 0.25 or 0.24999...
| 0.01 or 0.00111...
| 1/8 + 1/16 + 1/32 . . .
|-
| 1/5
| 0.2 or 0.1999...
| 0.00110011...
| 1/8 + 1/16 + 1/128 . . .
|-
| 1/6
| 0.1666...
| 0.0010101...
| 1/8 + 1/32 + 1/128 . . .
|-
| 1/7
| 0.142857142857...
| 0.001001...
| 1/8 + 1/64 + 1/512 . . .
|-
| 1/8
| 0.125 or 0.124999...
| 0.001 or 0.000111...
| 1/16 + 1/32 + 1/64 . . .
|-
| 1/9
| 0.111...
| 0.000111000111...
| 1/16 + 1/32 + 1/64 . . .
|-
| 1/10
| 0.1 or 0.0999...
| 0.000110011...
| 1/16 + 1/32 + 1/256 . . .
|-
| 1/11
| 0.090909...
| 0.00010111010001011101...
| 1/16 + 1/64 + 1/128 . . .
|-
| 1/12
| 0.08333...
| 0.00010101...
| 1/16 + 1/64 + 1/256 . . .
|-
| 1/13
| 0.076923076923...
| 0.000100111011000100111011...
| 1/16 + 1/128 + 1/256 . . .
|-
| 1/14
| 0.0714285714285...
| 0.0001001001...
| 1/16 + 1/128 + 1/1024 . . .
|-
| 1/15
| 0.0666...
| 0.00010001...
| 1/16 + 1/256 . . .
|-
| 1/16
| 0.0625 or 0.0624999...
| 0.0001 or 0.0000111...
| 1/32 + 1/64 + 1/128 . . .
|}

==Binary arithmetic==
Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.

===Addition===
[[Image:Half Adder.svg|thumbnail|200px|right|The circuit diagram for a binary half adder, which adds two bits together, producing sum and carry bits]]

The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:

:0 + 0 → 0
:0 + 1 → 1
:1 + 0 → 1
:1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21) )
Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:

:5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101) )
:7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101) )

This is known as ''carrying''. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:

1 1 1 1 1 (carried digits)
0 1 1 0 1
+ 1 0 1 1 1
-------------
= 1 0 0 1 0 0 = 36

In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 102. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 102 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 112. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).

When computers must add two numbers, the rule that:
x xor y = (x + y) mod 2
for any two bits x and y allows for very fast calculation, as well.

====Long carry method====
A simplification for many binary addition problems is the Long Carry Method or Brookhouse Method of Binary Addition. This method is generally useful in any binary addition in which one of the numbers contains a long "string" of ones. It is based on the simple premise that under the binary system, when given a "string" of digits composed entirely of ''n'' ones (where ''n'' is any integer length), adding 1 will result in the number 1 followed by a string of ''n'' zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of ''n'' 9s will result in the number 1 followed by a string of ''n'' 0s:

Binary Decimal
1 1 1 1 1 likewise 9 9 9 9 9
+ 1 + 1
——————————— ———————————
1 0 0 0 0 0 1 0 0 0 0 0

Such long strings are quite common in the binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations. In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 02 (95810) and 1 0 1 0 1 1 0 0 1 12 (69110), using the traditional carry method on the left, and the long carry method on the right:

Traditional Carry Method Long Carry Method
vs.
1 1 1 1 1 1 1 1 (carried digits) 1 ← 1 ← carry the 1 until it is one digit past the "string" below
1 1 1 0 1 1 1 1 1 0 <s>1 1 1</s> 0 <s>1 1 1 1 1</s> 0 cross out the "string",
+ 1 0 1 0 1 1 0 0 1 1 + 1 0 <s>1</s> 0 1 1 0 0 <s>1</s> 1 and cross out the digit that was added to it
——————————————————————— ——————————————————————
= 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1

The top row shows the carry bits used. Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series. The "used" numbers must be crossed off, since they are already added. Other long strings may likewise be cancelled using the same technique. Then, simply add together any remaining digits normally. Proceeding in this manner gives the final answer of 1 1 0 0 1 1 1 0 0 0 12 (164910). In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort.

====Addition table====

{| class="wikitable" style="text-align:center"
|-
! style="width:1.5em" |
! style="width:1.5em" | 0
! style="width:1.5em" | 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 10
|}

The binary addition table is similar, but not the same, as the truth table of the logical disjunction operation <math>\lor</math>. The difference is that <math>1</math><math>\lor </math><math>1=1</math>, while <math>1+1=10</math>.

=== Subtraction ===

Subtraction works in much the same way:

:0 − 0 → 0
:0 − 1 → 1, borrow 1
:1 − 0 → 1
:1 − 1 → 0
Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as ''borrowing''. The principle is the same as for carrying. When the result of a subtraction is less than 0, the least possible value of a digit, the procedure is to "borrow" the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value.

* * * * (starred columns are borrowed from)
1 1 0 1 1 1 0
− 1 0 1 1 1
----------------
= 1 0 1 0 1 1 1

* (starred columns are borrowed from)
1 0 1 1 1 1 1
- 1 0 1 0 1 1
----------------
= 0 1 1 0 1 0 0

Subtracting a positive number is equivalent to ''adding'' a negative number of equal absolute value. Computers use signed number representations to handle negative numbers—most commonly the two's complement notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula:

: {{math|1=A − B = A + not B + 1}}

===Multiplication===
Multiplication in binary is similar to its decimal counterpart. Two numbers ''A'' and ''B'' can be multiplied by partial products: for each digit in ''B'', the product of that digit in ''A'' is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in ''B'' that was used. The sum of all these partial products gives the final result.

Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:
* If the digit in ''B'' is 0, the partial product is also 0
* If the digit in ''B'' is 1, the partial product is equal to ''A''

For example, the binary numbers 1011 and 1010 are multiplied as follows:

1 0 1 1 (''A'')
× 1 0 1 0 (''B'')
---------
0 0 0 0 ← Corresponds to the rightmost 'zero' in ''B''
+ 1 0 1 1 ← Corresponds to the next 'one' in ''B''
+ 0 0 0 0
+ 1 0 1 1
---------------
= 1 1 0 1 1 1 0

Binary numbers can also be multiplied with bits after a binary point:

1 0 1 . 1 0 1 ''A'' (5.625 in decimal)
× 1 1 0 . 0 1 ''B'' (6.25 in decimal)
-------------------
1 . 0 1 1 0 1 ← Corresponds to a 'one' in ''B''
+ 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in ''B''
+ 0 0 0 . 0 0 0
+ 1 0 1 1 . 0 1
+ 1 0 1 1 0 . 1
---------------------------
= 1 0 0 0 1 1 . 0 0 1 0 1 (35.15625 in decimal)

See also Booth's multiplication algorithm.

====Multiplication table====
{| class="wikitable" style="text-align:center"
|-
! style="width:1.5em" |
! style="width:1.5em" | 0
! style="width:1.5em" | 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

The binary multiplication table is the same as the truth table of the logical conjunction operation <math>\land</math>.

===Division===

Long division in binary is again similar to its decimal counterpart.

In the example below, the divisor is 1012, or 5 in decimal, while the dividend is 110112, or 27 in decimal. The procedure is the same as that of decimal long division; here, the divisor 1012 goes into the first three digits 1102 of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a "1") is included to obtain a new three-digit sequence:

1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
0 0 1

The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:

1 0 1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
1 1 1
− 1 0 1
-----
0 1 0

Thus, the quotient of 110112 divided by 1012 is 1012, as shown on the top line, while the remainder, shown on the bottom line, is 102. In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2.

Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result.

===Square root===
The process of taking a binary square root digit by digit is the same as for a decimal square root and is explained here. An example is:

1 0 0 1
---------
√ 1010001
1
---------
101 01
0
--------
1001 100
0
--------
10001 10001
10001
-------
0

==Conversion to and from other numeral systems==

===Decimal to Binary===
[[File:Decimal to Binary Conversion.gif|alt=|frame|Conversion of (357)10 to binary notation results in (101100101)]]
To convert from a base-10 integer to its base-2 (binary) equivalent, the number is divided by two. The remainder is the least-significant bit. The quotient is again divided by two; its remainder becomes the next least significant bit. This process repeats until a quotient of one is reached. The sequence of remainders (including the final quotient of one) forms the binary value, as each remainder must be either zero or one when dividing by two. For example, (357)10 is expressed as (101100101)2.

=== Binary to Decimal ===
Conversion from base-2 to base-10 simply inverts the preceding algorithm. The bits of the binary number are used one by one, starting with the most significant (leftmost) bit. Beginning with the value 0, the prior value is doubled, and the next bit is then added to produce the next value. This can be organized in a multi-column table. For example, to convert 100101011012 to decimal:

{| style= "border: 1px solid #a2a9b1; border-spacing: 3px; background-color: #f8f9fa; color: black; margin: 0.5em 0 0.5em 1em; padding: 0.2em; line-height: 1.5em; width:22em"
!Prior value
! style="text-align:left" | × 2 +
!Next bit
!Next value
|-
|align="right"|0 ||× 2 +|| '''1''' || = 1
|-
|align="right"|1 ||× 2 +|| '''0''' || = 2
|-
|align="right"|2 ||× 2 +|| '''0''' || = 4
|-
|align="right"|4 ||× 2 +|| '''1''' || = 9
|-
|align="right"|9 ||× 2 +|| '''0''' || = 18
|-
|align="right"|18 ||× 2 +|| '''1''' || = 37
|-
|align="right"|37 ||× 2 +|| '''0''' || = 74
|-
|align="right"|74 ||× 2 +|| '''1''' || = 149
|-
|align="right"|149 ||× 2 +|| '''1''' || = 299
|-
|align="right"|299 ||× 2 +|| '''0''' || = 598
|-
|align="right"|598 ||× 2 +|| '''1''' || = '''1197'''
|}

The result is 119710. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the Horner scheme.

{|
! Binary 
| 1 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 || 0 || 1 ||
|-
! Decimal 
| 1×210 + || 0×29 + || 0×28 + || 1×27 + || 0×26 + || 1×25 + || 0×24 + || 1×23 + || 1×22 + || 0×21 + || 1×20 = || 1197
|}

The fractional parts of a number are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving.

In a fractional binary number such as 0.110101101012, the first digit is <math>\begin{matrix} \frac{1}{2} \end{matrix}</math>, the second <math>\begin{matrix} (\frac{1}{2})^2 = \frac{1}{4} \end{matrix}</math>, etc. So if there is a 1 in the first place after the decimal, then the number is at least <math>\begin{matrix} \frac{1}{2} \end{matrix}</math>, and vice versa. Double that number is at least 1. This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part.

For example, <math>\begin{matrix} (\frac{1}{3}) \end{matrix}</math>10, in binary, is:

{| class="wikitable"
!Converting!!Result
|-
|<math>\begin{matrix} \frac{1}{3} \end{matrix}</math> || 0.
|-
|<math>\begin{matrix} \frac{1}{3} \times 2 = \frac{2}{3} < 1 \end{matrix}</math> || 0.0
|-
|<math>\begin{matrix} \frac{2}{3} \times 2 = 1\frac{1}{3} \ge 1 \end{matrix}</math> || 0.01
|-
|<math>\begin{matrix} \frac{1}{3} \times 2 = \frac{2}{3} < 1 \end{matrix}</math> || 0.010
|-
|<math>\begin{matrix} \frac{2}{3} \times 2 = 1\frac{1}{3} \ge 1 \end{matrix}</math> || 0.0101
|}

Thus the repeating decimal fraction <math>0. \overline{3}</math>... is equivalent to the repeating binary fraction <math>0. \overline{01}</math>... .

Or for example, 0.110, in binary, is:

{| class="wikitable"
! Converting !! Result
|-
| '''0.1''' || 0.
|-
|0.1 × 2 = '''0.2''' < 1 || 0.0
|-
|0.2 × 2 = '''0.4''' < 1 || 0.00
|-
|0.4 × 2 = '''0.8''' < 1 || 0.000
|-
|0.8 × 2 = '''1.6''' ≥ 1 || 0.0001
|-
|0.6 × 2 = '''1.2''' ≥ 1 || 0.00011
|-
|0.2 × 2 = '''0.4''' < 1 || 0.000110
|-
|0.4 × 2 = '''0.8''' < 1 || 0.0001100
|-
|0.8 × 2 = '''1.6''' ≥ 1 || 0.00011001
|-
|0.6 × 2 = '''1.2''' ≥ 1 || 0.000110011
|-
|0.2 × 2 = '''0.4''' < 1 || 0.0001100110
|}

This is also a repeating binary fraction <math> 0.0 \overline{0011}</math>... . It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 0.1 + ... + 0.1, (10 additions) differs from 1 in floating point arithmetic. In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which 1/10 is not.

The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then divide by the appropriate power of two in the decimal base. For example:

: <math>
\begin{align}
x & = & 1100&.1\overline{01110}\ldots \\
x\times 2^6 & = & 1100101110&.\overline{01110}\ldots \\
x\times 2 & = & 11001&.\overline{01110}\ldots \\
x\times(2^6-2) & = & 1100010101 \\
x & = & 1100010101/111110 \\
x & = & (789/62)_{10}
\end{align}
</math>

Another way of converting from binary to decimal, often quicker for a person familiar with hexadecimal, is to do so indirectly—first converting (<math>x</math> in binary) into (<math>x</math> in hexadecimal) and then converting (<math>x</math> in hexadecimal) into (<math>x</math> in decimal).

For very large numbers, these simple methods are inefficient because they perform a large number of multiplications or divisions where one operand is very large. A simple divide-and-conquer algorithm is more effective asymptotically: given a binary number, it is divided by 10''k'', where ''k'' is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two are concatenated. Given a decimal number, it can be split into two pieces of about the same size, each of which is converted to binary, whereupon the first converted piece is multiplied by 10''k'' and added to the second converted piece, where ''k'' is the number of decimal digits in the second, least-significant piece before conversion.

===Hexadecimal===
Binary may be converted to and from hexadecimal more easily. This is because the radix of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 24, so it takes four digits of binary to represent one digit of hexadecimal, as shown in the adjacent table.

To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:

:3A16 = 0011 10102
:E716 = 1110 01112

To convert a binary number into its hexadecimal equivalent, divide it into groups of four bits. If the number of bits isn't a multiple of four, simply insert extra '''0''' bits at the left (called padding). For example:

:10100102 = 0101 0010 grouped with padding = 5216
:110111012 = 1101 1101 grouped = DD16

To convert a hexadecimal number into its decimal equivalent, multiply the decimal equivalent of each hexadecimal digit by the corresponding power of 16 and add the resulting values:

:C0E716 = (12 × 163) + (0 × 162) + (14 × 161) + (7 × 160) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,38310

===Octal===
Binary is also easily converted to the octal numeral system, since octal uses a radix of 8, which is a power of two (namely, 23, so it takes exactly three binary digits to represent an octal digit). The correspondence between octal and binary numerals is the same as for the first eight digits of hexadecimal in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth.

{| class="wikitable" style="text-align:center"
!Octal!!Binary
|-
| 0 || 000
|-
| 1 || 001
|-
| 2 || 010
|-
| 3 || 011
|-
| 4 || 100
|-
| 5 || 101
|-
| 6 || 110
|-
| 7 || 111
|}

Converting from octal to binary proceeds in the same fashion as it does for hexadecimal:

:658 = 110 1012
:178 = 001 1112

And from binary to octal:

:1011002 = 101 1002 grouped = 548
:100112 = 010 0112 grouped with padding = 238

And from octal to decimal:

:658 = (6 × 81) + (5 × 80) = (6 × 8) + (5 × 1) = 5310
:1278 = (1 × 82) + (2 × 81) + (7 × 80) = (1 × 64) + (2 × 8) + (7 × 1) = 8710

==Representing real numbers==

Non-integers can be represented by using negative powers, which are set off from the other digits by means of a radix point (called a decimal point in the decimal system). For example, the binary number 11.012 means:

{|
|'''1''' × 21 || (1 × 2 = '''2''') || plus
|-
|'''1''' × 20 || (1 × 1 = '''1''') || plus
|-
|'''0''' × 2−1 || (0 × <math> \tfrac{1}{2} </math> = '''0''') || plus
|-
|'''1''' × 2−2 || (1 × <math> \tfrac{1}{4} </math> = '''0.25''')
|}

For a total of 3.25 decimal.

All dyadic rational numbers <math>\frac{p}{2^a}</math> have a ''terminating'' binary numeral—the binary representation has a finite number of terms after the radix point. Other rational numbers have binary representation, but instead of terminating, they ''recur'', with a finite sequence of digits repeating indefinitely. For instance

:<math>\frac{1_{10}}{3_{10}} = \frac{1_2}{11_2} = 0.01010101\overline{01}\ldots\,_2 </math>

:<math>\frac{12_{10}}{17_{10}} = \frac{1100_2}{10001_2} = 0.10110100 10110100\overline{10110100}\ldots\,_2 </math>

The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in decimal. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the geometric series 2−1 + 2−2 + 2−3 + ... which is 1.

Binary numerals which neither terminate nor recur represent irrational numbers. For instance,
* 0.10100100010000100000100... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
* 1.0110101000001001111001100110011111110... is the binary representation of <math>\sqrt{2}</math>, the square root of 2, another irrational. It has no discernible pattern.

= Base 5 =
'''Quinary''' /ˈkwaɪnəri/ ('''base-5''' or '''pental''') is a numeral system with five as the base. A possible origination of a quinary system is that there are five digits on either hand.

In the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100 and sixty is written as 220.

As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two highly composite numbers (4 and 6) guarantees that many recurring fractions have relatively short periods.

Today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a sub-base system, is sexagesimal, base 60, which used 10 as a sub-base.

Each quinary digit can hold log25 (approx. 2.32) bits of information.

==Comparison to other radices==
{| class="wikitable" style="float:right; text-align:center"
|+ A quinary multiplication table
|-
| × || '''1''' || '''2''' || '''3''' || '''4''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''20'''
|-
| '''1''' || 1 || 2 || 3 || 4 || 10 || 11 || 12 || 13 || 14 || 20
|-
| '''2''' || 2 || 4 || 11 || 13 || 20 || 22 || 24 || 31 || 33 || 40
|-
| '''3''' || 3 || 11 || 14 || 22 || 30 || 33 || 41 || 44 || 102 || 110
|-
| '''4''' || 4 || 13 || 22 || 31 || 40 || 44 || 103 || 112 || 121 || 130
|-
| '''10''' || 10 || 20 || 30 || 40 || 100 || 110 || 120 || 130 || 140 || 200
|-
| '''11''' || 11 || 22 || 33 || 44 || 110 || 121 || 132 || 143 || 204 || 220
|-
| '''12''' || 12 || 24 || 41 || 103 || 120 || 132 || 144 || 211 || 223 || 240
|-
| '''13''' || 13 || 31 || 44 || 112 || 130 || 143 || 211 || 224 || 242 || 310
|-
| '''14''' || 14 || 33 || 102 || 121 || 140 || 204 || 223 || 242 || 311 || 330
|-
| '''20''' || 20 || 40 || 110 || 130 || 200 || 220 || 240 || 310 || 330 || 400
|}

{| class="wikitable"
|+ '''Numbers zero to twenty-five in standard quinary'''
|- align="center"
! Quinary
| 0 || 1 || 2 || 3 || 4 || 10 || 11 || 12 || 13 || 14 || 20 || 21 || 22
|- align="center"
! Binary
| 0 || 1 || 10 || 11 || 100 || 101 || 110 || 111 || 1000 || 1001 || 1010 || 1011 || 1100
|- align="center"
! Decimal
! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12
|- align="center"
!
|- align="center"
! Quinary
| 23 || 24 || 30 || 31 || 32 || 33 || 34 || 40 || 41 || 42 || 43 || 44 || 100
|- align="center"
! Binary
| 1101 || 1110 || 1111 || 10000 || 10001 || 10010 || 10011 || 10100 || 10101 || 10110 || 10111 || 11000 || 11001
|- align="center"
! Decimal
! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20 !! 21 !! 22 !! 23 !! 24 !! 25
|}

{| class="wikitable"
|+ '''Fractions in quinary'''
|'''Decimal''' (periodic part) ||'''Quinary''' (periodic part)
|'''Binary''' (periodic part)
|-
|1/2 = 0.5
|'''1/2''' = 0.2
|1/10 = 0.1
|-
|1/3 = 0.3
|'''1/3''' = 0.13
|1/11 = 0.01
|-
|1/4 = 0.25
|'''1/4''' = 0.1
|1/100 = 0.01
|-
|1/5 = 0.2
|'''1/10''' = 0.1
|1/101 = 0.0011
|-
|1/6 = 0.16
|'''1/11''' = 0.04
|1/110 = 0.010
|-
|1/7 = 0.142857
|'''1/12''' = 0.032412
|1/111 = 0.001
|-
|1/8 = 0.125
|'''1/13''' = 0.03
|1/1000 = 0.001
|-
|1/9 = 0.1
|'''1/14''' = 0.023421
|1/1001 = 0.000111
|-
|1/10 = 0.1
|'''1/20''' = 0.02
|1/1010 = 0.00011
|-
|1/11 = 0.09
|'''1/21''' = 0.02114
|1/1011 = 0.0001011101
|-
|1/12 = 0.083
|'''1/22''' = 0.02
|1/1100 = 0.0001
|-
|1/13 = 0.076923
|'''1/23''' = 0.0143
|1/1101 = 0.000100111011
|-
|1/14 = 0.0714285
|'''1/24''' = 0.013431
|1/1110 = 0.0001
|-
|1/15 = 0.06
|'''1/30''' = 0.013
|1/1111 = 0.0001
|-
|1/16 = 0.0625
|'''1/31''' = 0.0124
|1/10000 = 0.0001
|-
|1/17 = 0.0588235294117647
|'''1/32''' = 0.0121340243231042
|1/10001 = 0.00001111
|-
|1/18 = 0.05
|'''1/33''' = 0.011433
|1/10010 = 0.0000111
|-
|1/19 = 0.052631578947368421
|'''1/34''' = 0.011242141
|1/10011 = 0.000011010111100101
|-
|1/20 = 0.05
|'''1/40''' = 0.01
|1/10100 = 0.000011
|-
|1/21 = 0.047619
|'''1/41''' = 0.010434
|1/10101 = 0.000011
|-
|1/22 = 0.045
|'''1/42''' = 0.01032
|1/10110 = 0.00001011101
|-
|1/23 = 0.0434782608695652173913
|'''1/43''' = 0.0102041332143424031123
|1/10111 = 0.00001011001
|-
|1/24 = 0.0416
|'''1/44''' = 0.01
|1/11000 = 0.00001
|-
|1/25 = 0.04
|'''1/100''' = 0.01
|1/11001 = 0.00001010001111010111
|}

==Usage==
Many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below:

{| class="wikitable" border="1" style="text-align:center"
! Number !! Base 5 !! Numeral
|-
! style="text-align:right" | 1
| 1
| wanggany
|-
! style="text-align:right" | 2
| 2
| marrma
|-
! style="text-align:right" | 3
| 3
| lurrkun
|-
! style="text-align:right" | 4
| 4
| dambumiriw
|-
! style="text-align:right" | 5
| 10
| wanggany rulu
|-
! style="text-align:right" | 10
| 20
| marrma rulu
|-
! style="text-align:right" | 15
| 30
| lurrkun rulu
|-
! style="text-align:right" | 20
| 40
| dambumiriw rulu
|-
! style="text-align:right" | 25
| 100
| dambumirri rulu
|-
! style="text-align:right" | 50
| 200
| marrma dambumirri rulu
|-
! style="text-align:right" | 75
| 300
| lurrkun dambumirri rulu
|-
! style="text-align:right" | 100
| 400
| dambumiriw dambumirri rulu
|-
! style="text-align:right" | 125
| 1000
| dambumirri dambumirri rulu
|-
! style="text-align:right" | 625
| 10000

| dambumirri dambumirri dambumirri rulu
|}

==Biquinary==
[[File:Chinese-abacus.jpg|thumb|right|Chinese Abacus or suanpan]]
A decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are an early biquinary system. The numbers 1, 5, 10, and 50 are written as '''I''', '''V''', '''X''', and '''L''' respectively. Seven is '''VII''' and seventy is '''LXX'''. The full list is:
{| class="wikitable" style="text-align:center"
| '''I''' || '''V''' || '''X''' || '''L''' || '''C''' || '''D''' || '''M'''
|-
| 1 || 5 || 10 || 50 || 100 || 500 || 1000
|}
Note that these are not positional number systems. In theory a number such as 73 could be written as IIIXXL without ambiguity as well as LXXIII and it is still not possible to extend it beyond thousands. There is also no sign for zero. But with the introduction of inversions such as IV and IX, it was necessary to keep the order from most to least significant.

Many versions of the abacus, such as the suanpan and soroban, use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary. Units of currencies are commonly partially or wholly biquinary.

Bi-quinary coded decimal is a variant of biquinary that was used on a number of early computers including Colossus and the IBM 650 to represent decimal numbers.

= Base 10 =

The '''decimal''' numeral system (also called the '''base-ten''' positional numeral system, and occasionally called '''denary''' /ˈdiːnəri/ or '''decanary''') is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''.

A ''decimal numeral'' (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in 25.9703 or 3,1415). ''Decimal'' may also refer specifically to the digits after the decimal separator, such as in "3.14 is the approximation of <math> \pi </math> to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.

The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form {{math|''a''/10''n''}}, where {{math|''a''}} is an integer, and {{math|''n''}} is a non-negative integer.

The decimal system has been extended to ''infinite decimals'' for representing any real number, by using an infinite sequence of digits after the decimal separator (see decimal representation). In this context, the decimal numerals with a finite number of non-zero digits after the decimal separator are sometimes called ''terminating decimals''. A ''repeating decimal'' is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., 5.123144144144144... = <math> 5.123 \overline{144}</math>). An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

==Origin==
[[File:Two hand, ten fingers.jpg|thumb|right|Ten fingers on two hands, the possible origin of decimal counting]]
Many numeral systems of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the Egyptian numerals, then the Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called ''decimal fractions'' or ''decimal numbers'', for forming the ''decimal numeral system''.

== Decimal notation ==
For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "−". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; and a comma "{{math|,}}" in other countries.

For representing a non-negative number, a decimal numeral consists of
* either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer,
*:<math>a_ma_{m-1}\ldots a_0</math>
*or a decimal mark separating two sequences of digits (such as "20.70828")
::<math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math>.
If {{math|''m'' > 0}}, that is, if the first sequence contains at least two digits, it is generally assumed that the first digit {{math|''a''''m''}} is not zero. In some circumstances it may be useful to have one or more 0's on the left; this does not change the value represented by the decimal: for example, {{math|1=3.14 = 03.14 = 003.14}}. Similarly, if the final digit on the right of the decimal mark is zero—that is, if {{math|1=''b''''n'' = 0}}—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; for example, {{math|1=15 = 15.0 = 15.00}} and {{math|1=5.2 = 5.20 = 5.200}}.

For representing a negative number, a minus sign is placed before {{math|''a''''m''}}.

The numeral <math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math> represents the number
:<math>a_m10^m+a_{m-1}10^{m-1}+\cdots+a_{0}10^0+\frac{b_1}{10^1}+\frac{b_2}{10^2}+\cdots+\frac{b_n}{10^n}</math>.
The ''integer part'' or ''integral part'' of a decimal numeral is the integer written to the left of the decimal separator (see also truncation). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the ''fractional part'', which equals the difference between the numeral and its integer part.

When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example {{math|.1234}}, instead of {{math|0.1234}}). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation.

In brief, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is a positional numeral system.

== Decimal fractions ==
'''Decimal fractions''' (sometimes called '''decimal numbers''', especially in contexts involving explicit fractions) are the rational numbers that may be expressed as a fraction whose denominator is a power of ten. For example, the decimals <math>0.8, 14.89, 0.00024, 1.618, 3.14159</math> represent the fractions <math> \tfrac{8}{10}</math>, <math> \tfrac{1489}{100}</math>, <math> \tfrac{24}{100000}</math>, <math>\tfrac {1618}{1000}</math> and <math>\tfrac{314159}{100000}</math>, and are therefore decimal numbers.

More generally, a decimal with {{math|''n''}} digits after the separator represents the fraction with denominator {{math|10''n''}}, whose numerator is the integer obtained by removing the separator.

It follows that a number is a decimal fraction if and only if it has a finite decimal representation.

Expressed as a fully reduced fraction, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are
:<math>1=2^0\cdot 5^0, 2=2^1\cdot 5^0, 4=2^2\cdot 5^0, 5=2^0\cdot 5^1, 8=2^3\cdot 5^0, 10=2^1\cdot 5^1, 16=2^4\cdot 5^0, 25=2^0\cdot 5^2, \ldots</math>

==Real number approximation==

Decimal numerals do not allow an exact representation for all real numbers, e.g. for the real number <math>\pi</math>. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates the real <math>\pi</math>, being less than 10−5 off; so decimals are widely used in science, engineering and everyday life.

More precisely, for every real number {{Mvar|x}} and every positive integer {{Mvar|n}}, there are two decimals {{Mvar|''L''}} and {{Mvar|''u''}} with at most ''{{Mvar|n}}'' digits after the decimal mark such that {{Math|''L'' ≤ ''x'' ≤ ''u''}} and {{Math|1=(''u'' − ''L'') = 10−''n''}}.

Numbers are very often obtained as the result of measurement. As measurements are subject to measurement uncertainty with a known upper bound, the result of a measurement is well-represented by a decimal with {{math|''n''}} digits after the decimal mark, as soon as the absolute measurement error is bounded from above by {{Math|10−''n''}}. In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also significant figures).

==Infinite decimal expansion==

For a real number {{Mvar|x}} and an integer {{Math|''n'' ≥ 0}}, let {{Math|[''x'']''n''}} denote the (finite) decimal expansion of the greatest number that is not greater than ''{{Mvar|x}}'' that has exactly {{Mvar|n}} digits after the decimal mark. Let {{Math|''d''''i''}} denote the last digit of {{Math|[''x'']''i''}}. It is straightforward to see that {{Math|[''x'']''n''}} may be obtained by appending {{Math|''d''''n''}} to the right of {{Math|[''x'']''n''−1}}. This way one has
:{{Math|1=[''x'']''n'' = [''x'']0.''d''1''d''2...''d''''n''−1''d''''n''}},

and the difference of {{Math|[''x'']''n''−1}} and {{Math|[''x'']''n''}} amounts to
:<math>\left\vert \left [ x \right ]_n-\left [ x \right ]_{n-1} \right\vert=d_n\cdot10^{-n}<10^{-n+1}</math>,

which is either 0, if {{Math|1=''d''''n'' = 0}}, or gets arbitrarily small as ''{{Mvar|n}}'' tends to infinity. According to the definition of a limit, ''{{Mvar|x}}'' is the limit of {{Math|[''x'']''n''}} when ''{{Mvar|n}}'' tends to infinity. This is written as<math display="inline">\; x = \lim_{n\rightarrow\infty} [x]_n \;</math>or
: {{Math|1=''x'' = [''x'']0.''d''1''d''2...''d''''n''...}},
which is called an '''infinite decimal expansion''' of ''{{Mvar|x}}''.

Conversely, for any integer {{Math|[''x'']0}} and any sequence of digits<math display="inline">\;(d_n)_{n=1}^{\infty}</math> the (infinite) expression {{Math|[''x'']0.''d''1''d''2...''d''''n''...}} is an ''infinite decimal expansion'' of a real number ''{{Mvar|x}}''. This expansion is unique if neither all {{Math|''d''''n''}} are equal to 9 nor all {{Math|''d''''n''}} are equal to 0 for ''{{Mvar|n}}'' large enough (for all ''{{Mvar|n}}'' greater than some natural number {{Mvar|N}}).

If all {{Math|''d''''n''}} for {{Math|''n'' > ''N''}} equal to 9 and {{Math|1=[''x'']''n'' = [''x'']0.''d''1''d''2...''d''''n''}}, the limit of the sequence<math display="inline">\;([x]_n)_{n=1}^{\infty}</math> is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: {{Math|''d''''N''}}, by {{Math|''d''''N'' + 1}}, and replacing all subsequent 9s by 0s (see 0.999...).

Any such decimal fraction, i.e.: {{Math|1=''d''''n'' = 0}} for {{Math|''n'' > ''N''}}, may be converted to its equivalent infinite decimal expansion by replacing {{Math|''d''''N''}} by {{Math|''d''''N'' − 1}} and replacing all subsequent 0s by 9s (see 0.999...).

In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of {{Math|[''x'']''n''}}, and the other containing only 9s after some place, which is obtained by defining {{Math|[''x'']''n''}} as the greatest number that is ''less'' than {{Mvar|x}}, having exactly ''{{Mvar|n}}'' digits after the decimal mark.

=== Rational numbers ===

Long division allows computing the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a ''repeating decimal''. For example,
:<math> \tfrac{1}{81}</math> = 0. 012345679 012... (with the group 012345679 indefinitely repeating).

The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational.
{|
|-
|For example, if ''x'' is || 0.4156156156...
|-
|then 10,000''x'' is || 4156.156156156...
|-
|and 10''x'' is|| 4.156156156...
|-
|so 10,000''x'' − 10''x'', i.e. 9,990''x'', is|| 4152.000000000...
|-
|and ''x'' is|| <math> \tfrac{4152}{9990}</math>
|}
or, dividing both numerator and denominator by 6, <math> \tfrac{692}{1665}</math>.

== Decimal computation ==

[[File:Decimal multiplication table.JPG|thumb|right|300px|Diagram of the world's earliest known multiplication table (c. 305 BCE) from the Warring States period]]

Most modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as the ENIAC or the IBM 650, used decimal representation internally).
For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems.

For most purposes, however, binary values are converted to or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)

Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of binary-coded decimal, especially in database implementations, but there are other decimal representations in use (including decimal floating point such as in newer revisions of the IEEE 754|IEEE 754 Standard for Floating-Point Arithmetic).

Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of <math>10</math> have no finite binary fractional representation; and is generally impossible for multiplication (or division). See Arbitrary-precision arithmetic for exact calculations.

= Licensing =
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Radix Radix, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Binary_number Binary number, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Quinary Quinary, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Decimal Decimal, Wikipedia] under a CC BY-SA license

Base 10, Base 2 & Base 5

2022-02-05T19:08:40Z

Khanh: /* Conversion to and from other numeral systems */

In a positional numeral system, the '''radix''' or '''base''' is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9.

In any standard positional numeral system, a number is conventionally written as (''x'')''y'' with ''x'' as the string of digits and ''y'' as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)10 is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.

= In numeral systems =
In the system with radix 13, for example, a string of digits such as 398 denotes the (decimal) number 3 × 132 + 9 × 131 + 8 × 130 = 632.

More generally, in a system with radix ''b'' (''b'' > 1), a string of digits ''d''1 … ''dn'' denotes the number ''d''1''b''''n''−1 + ''d''2''b''''n''−2 + … + ''dnb''0, where 0 ≤ ''di'' < ''b''. In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radix ''b'' would have a ones' place, then a ''b''1s' place, a ''b''2s' place, etc.

Commonly used numeral systems include:
{| class="wikitable sortable"
! Base/radix
! Name
! Description
|-
| 2
| Binary numeral system
| Used internally by nearly all computers, is base 2. The two digits are "0" and "1", expressed from switches displaying OFF and ON, respectively. Used in most electric counters.
|-
| 8
| Octal system
| Used occasionally in computing. The eight digits are "0"–"7" and represent 3 bits (23).
|-
| 10
| Decimal system
| Used by humans in the vast majority of cultures. Its ten digits are "0"–"9". Used in most mechanical counters.
|-
| 12
| Duodecimal (dozenal) system
| Sometimes advocated due to divisibility by 2, 3, 4, and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
|-
| 16
| Hexadecimal system
| Often used in computing as a more compact representation of binary (1 hex digit per 4 bits). The sixteen digits are "0"–"9" followed by "A"–"F" or "a"–"f".
|-
| 20
| Vigesimal system
| Traditional numeral system in several cultures, still used by some for counting. Historically also known as the ''score system'' in English, now most famous in the phrase "four score and seven years ago" in the Gettysburg Address.
|-
| 60
| Sexagesimal system
| Originated in ancient Sumer and passed to the Babylonians. Used today as the basis of modern circular coordinate system (degrees, minutes, and seconds) and time measuring (minutes, and seconds) by analogy to the rotation of the Earth.
|}

= Base 2 =
A '''binary number''' is a number expressed in the '''base-2 numeral system''' or '''binary numeral system''', a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).

The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language.

==Representation==
Any number can be represented by a sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. Any of the following rows of symbols can be interpreted as the binary numeric value of 667:

{| style="text-align:center;"
| 1 || 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1 || 1
|-
| <nowiki>|</nowiki> || ― || <nowiki>|</nowiki> || ― || ― || <nowiki>|</nowiki> || <nowiki>|</nowiki> || ― || <nowiki>|</nowiki> || <nowiki>|</nowiki>
|-
| ☒ || ☐ || ☒ || ☐ || ☐ || ☒ || ☒ || ☐ || ☒ || ☒
|-
| y || n || y || n || n || y || y || n || y || y
|}

[[Image:Binary clock.svg|250px|thumbnail|right|A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.]]

The numeric value represented in each case is dependent upon the value assigned to each symbol. In the earlier days of computing, switches, punched holes and punched paper tapes were used to represent binary values. In a modern computer, the numeric values may be represented by two different voltages; on a magnetic disk, magnetic polarities may be used. A "positive", "yes", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.

In keeping with customary representation of numerals using Arabic numerals, binary numbers are commonly written using the symbols '''0''' and '''1'''. When written, binary numerals are often subscripted, prefixed or suffixed in order to indicate their base, or radix. The following notations are equivalent:
* 100101 binary (explicit statement of format)
* 100101b (a suffix indicating binary format; also known as Intel convention)
* 100101B (a suffix indicating binary format)
* bin 100101 (a prefix indicating binary format)
* 1001012 (a subscript indicating base-2 (binary) notation)
* %100101 (a prefix indicating binary format; also known as Motorola convention)
* 0b100101 (a prefix indicating binary format, common in programming languages)
* 6b100101 (a prefix indicating number of bits in binary format, common in programming languages)
* #b100101 (a prefix indicating binary format, common in Lisp programming languages)

When spoken, binary numerals are usually read digit-by-digit, in order to distinguish them from decimal numerals. For example, the binary numeral 100 is pronounced ''one zero zero'', rather than ''one hundred'', to make its binary nature explicit, and for purposes of correctness. Since the binary numeral 100 represents the value four, it would be confusing to refer to the numeral as ''one hundred'' (a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct ''value''), but this does not make its binary nature explicit.

==Counting in binary==
Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar decimal counting system as a frame of reference.

===Decimal counting===
Decimal counting uses the ten symbols ''0'' through ''9''. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the ''first digit''. When the available symbols for this position are exhausted, the least significant digit is reset to ''0'', and the next digit of higher significance (one position to the left) is incremented (''overflow''), and incremental substitution of the low-order digit resumes. This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows:

:000, 001, 002, ... 007, 008, 009, (rightmost digit is reset to zero, and the digit to its left is incremented)
:0'''1'''0, 011, 012, ...
:   ...
:090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented)
:'''1'''00, 101, 102, ...

===Binary counting===
[[File:Binary counter.gif|thumb|This counter shows how to count in binary from numbers zero through thirty-one.]]
[[File:Binary_guess_number_trick_SMIL.svg|thumb|upright|link={{filepath:binary_guess_number_trick_SMIL.svg}}|A party trick to guess a number from which cards it is printed on uses the bits of the binary representation of the number. In the SVG file, click a card to toggle it]]
Binary counting follows the same procedure, except that only the two symbols ''0'' and ''1'' are available. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:

:0000,
:000'''1''', (rightmost digit starts over, and next digit is incremented)
:00'''1'''0, 0011, (rightmost two digits start over, and next digit is incremented)
:0'''1'''00, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented)
:'''1'''000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...

In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" digit. For example, the binary number 100101 is converted to decimal form as follows:

:1001012 = [ ( '''1''' ) × 25 ] + [ ( '''0''' ) × 24 ] + [ ( '''0''' ) × 23 ] + [ ( '''1''' ) × 22 ] + [ ( '''0''' ) × 21 ] + [ ( '''1''' ) × 20 ]

:1001012 = [ '''1''' × 32 ] + [ '''0''' × 16 ] + [ '''0''' × 8 ] + [ '''1''' × 4 ] + [ '''0''' × 2 ] + [ '''1''' × 1 ]

:'''1001012 = 3710'''

==Fractions==

Fractions in binary arithmetic terminate only if 2 is the only prime factor in the denominator. As a result, 1/10 does not have a finite binary representation ('''10''' has prime factors '''2''' and '''5'''). This causes 10 × 0.1 not to precisely equal 1 in floating-point arithmetic. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × '''2−1''' + 1 × '''2−2''' + 0 × '''2−3''' + 1 × '''2−4''' + ... = 0.3125 + ... An exact value cannot be found with a sum of a finite number of inverse powers of two, the zeros and ones in the binary representation of 1/3 alternate forever.

{| class="wikitable"
|-
! Fraction
! [[Base 10|Decimal]]
! Binary
! Fractional approximation
|-
| 1/1
| 1{{pad|0.25em}}or{{pad|0.25em}}0.999...
| 1{{pad|0.25em}}or{{pad|0.25em}}0.111...
| 1/2 + 1/4 + 1/8...
|-
| 1/2
| 0.5{{pad|0.25em}}or{{pad|0.25em}}0.4999...
| 0.1{{pad|0.25em}}or{{pad|0.25em}}0.0111...
| 1/4 + 1/8 + 1/16 . . .
|-
| 1/3
| 0.333...
| 0.010101...
| 1/4 + 1/16 + 1/64 . . .
|-
| 1/4
| 0.25{{pad|0.25em}}or{{pad|0.25em}}0.24999...
| 0.01{{pad|0.25em}}or{{pad|0.25em}}0.00111...
| 1/8 + 1/16 + 1/32 . . .
|-
| 1/5
| 0.2{{pad|0.25em}}or{{pad|0.25em}}0.1999...
| 0.00110011...
| 1/8 + 1/16 + 1/128 . . .
|-
| 1/6
| 0.1666...
| 0.0010101...
| 1/8 + 1/32 + 1/128 . . .
|-
| 1/7
| 0.142857142857...
| 0.001001...
| 1/8 + 1/64 + 1/512 . . .
|-
| 1/8
| 0.125{{pad|0.25em}}or{{pad|0.25em}}0.124999...
| 0.001{{pad|0.25em}}or{{pad|0.25em}}0.000111...
| 1/16 + 1/32 + 1/64 . . .
|-
| 1/9
| 0.111...
| 0.000111000111...
| 1/16 + 1/32 + 1/64 . . .
|-
| 1/10
| 0.1{{pad|0.25em}}or{{pad|0.25em}}0.0999...
| 0.000110011...
| 1/16 + 1/32 + 1/256 . . .
|-
| 1/11
| 0.090909...
| 0.00010111010001011101...
| 1/16 + 1/64 + 1/128 . . .
|-
| 1/12
| 0.08333...
| 0.00010101...
| 1/16 + 1/64 + 1/256 . . .
|-
| 1/13
| 0.076923076923...
| 0.000100111011000100111011...
| 1/16 + 1/128 + 1/256 . . .
|-
| 1/14
| 0.0714285714285...
| 0.0001001001...
| 1/16 + 1/128 + 1/1024 . . .
|-
| 1/15
| 0.0666...
| 0.00010001...
| 1/16 + 1/256 . . .
|-
| 1/16
| 0.0625{{pad|0.25em}}or{{pad|0.25em}}0.0624999...
| 0.0001{{pad|0.25em}}or{{pad|0.25em}}0.0000111...
| 1/32 + 1/64 + 1/128 . . .
|}

==Binary arithmetic==
Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.

===Addition===
[[Image:Half Adder.svg|thumbnail|200px|right|The circuit diagram for a binary half adder, which adds two bits together, producing sum and carry bits]]

The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:

:0 + 0 → 0
:0 + 1 → 1
:1 + 0 → 1
:1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21) )
Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:

:5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101) )
:7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101) )

This is known as ''carrying''. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:

1 1 1 1 1 (carried digits)
0 1 1 0 1
+ 1 0 1 1 1
-------------
= 1 0 0 1 0 0 = 36

In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 102. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 102 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 112. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).

When computers must add two numbers, the rule that:
x xor y = (x + y) mod 2
for any two bits x and y allows for very fast calculation, as well.

====Long carry method====
A simplification for many binary addition problems is the Long Carry Method or Brookhouse Method of Binary Addition. This method is generally useful in any binary addition in which one of the numbers contains a long "string" of ones. It is based on the simple premise that under the binary system, when given a "string" of digits composed entirely of ''n'' ones (where ''n'' is any integer length), adding 1 will result in the number 1 followed by a string of ''n'' zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of ''n'' 9s will result in the number 1 followed by a string of ''n'' 0s:

Binary Decimal
1 1 1 1 1 likewise 9 9 9 9 9
+ 1 + 1
——————————— ———————————
1 0 0 0 0 0 1 0 0 0 0 0

Such long strings are quite common in the binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations. In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 02 (95810) and 1 0 1 0 1 1 0 0 1 12 (69110), using the traditional carry method on the left, and the long carry method on the right:

Traditional Carry Method Long Carry Method
vs.
1 1 1 1 1 1 1 1 (carried digits) 1 ← 1 ← carry the 1 until it is one digit past the "string" below
1 1 1 0 1 1 1 1 1 0 <s>1 1 1</s> 0 <s>1 1 1 1 1</s> 0 cross out the "string",
+ 1 0 1 0 1 1 0 0 1 1 + 1 0 <s>1</s> 0 1 1 0 0 <s>1</s> 1 and cross out the digit that was added to it
——————————————————————— ——————————————————————
= 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1

The top row shows the carry bits used. Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series. The "used" numbers must be crossed off, since they are already added. Other long strings may likewise be cancelled using the same technique. Then, simply add together any remaining digits normally. Proceeding in this manner gives the final answer of 1 1 0 0 1 1 1 0 0 0 12 (164910). In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort.

====Addition table====

{| class="wikitable" style="text-align:center"
|-
! style="width:1.5em" |
! style="width:1.5em" | 0
! style="width:1.5em" | 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 10
|}

The binary addition table is similar, but not the same, as the truth table of the logical disjunction operation <math>\lor</math>. The difference is that <math>1</math><math>\lor </math><math>1=1</math>, while <math>1+1=10</math>.

=== Subtraction ===

Subtraction works in much the same way:

:0 − 0 → 0
:0 − 1 → 1, borrow 1
:1 − 0 → 1
:1 − 1 → 0
Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as ''borrowing''. The principle is the same as for carrying. When the result of a subtraction is less than 0, the least possible value of a digit, the procedure is to "borrow" the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value.

* * * * (starred columns are borrowed from)
1 1 0 1 1 1 0
− 1 0 1 1 1
----------------
= 1 0 1 0 1 1 1

* (starred columns are borrowed from)
1 0 1 1 1 1 1
- 1 0 1 0 1 1
----------------
= 0 1 1 0 1 0 0

Subtracting a positive number is equivalent to ''adding'' a negative number of equal absolute value. Computers use signed number representations to handle negative numbers—most commonly the two's complement notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula:

: {{math|1=A − B = A + not B + 1}}

===Multiplication===
Multiplication in binary is similar to its decimal counterpart. Two numbers ''A'' and ''B'' can be multiplied by partial products: for each digit in ''B'', the product of that digit in ''A'' is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in ''B'' that was used. The sum of all these partial products gives the final result.

Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:
* If the digit in ''B'' is 0, the partial product is also 0
* If the digit in ''B'' is 1, the partial product is equal to ''A''

For example, the binary numbers 1011 and 1010 are multiplied as follows:

1 0 1 1 (''A'')
× 1 0 1 0 (''B'')
---------
0 0 0 0 ← Corresponds to the rightmost 'zero' in ''B''
+ 1 0 1 1 ← Corresponds to the next 'one' in ''B''
+ 0 0 0 0
+ 1 0 1 1
---------------
= 1 1 0 1 1 1 0

Binary numbers can also be multiplied with bits after a binary point:

1 0 1 . 1 0 1 ''A'' (5.625 in decimal)
× 1 1 0 . 0 1 ''B'' (6.25 in decimal)
-------------------
1 . 0 1 1 0 1 ← Corresponds to a 'one' in ''B''
+ 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in ''B''
+ 0 0 0 . 0 0 0
+ 1 0 1 1 . 0 1
+ 1 0 1 1 0 . 1
---------------------------
= 1 0 0 0 1 1 . 0 0 1 0 1 (35.15625 in decimal)

See also Booth's multiplication algorithm.

====Multiplication table====
{| class="wikitable" style="text-align:center"
|-
! style="width:1.5em" |
! style="width:1.5em" | 0
! style="width:1.5em" | 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

The binary multiplication table is the same as the truth table of the logical conjunction operation <math>\land</math>.

===Division===

Long division in binary is again similar to its decimal counterpart.

In the example below, the divisor is 1012, or 5 in decimal, while the dividend is 110112, or 27 in decimal. The procedure is the same as that of decimal long division; here, the divisor 1012 goes into the first three digits 1102 of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a "1") is included to obtain a new three-digit sequence:

1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
0 0 1

The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:

1 0 1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
1 1 1
− 1 0 1
-----
0 1 0

Thus, the quotient of 110112 divided by 1012 is 1012, as shown on the top line, while the remainder, shown on the bottom line, is 102. In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2.

Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result.

===Square root===
The process of taking a binary square root digit by digit is the same as for a decimal square root and is explained here. An example is:

1 0 0 1
---------
√ 1010001
1
---------
101 01
0
--------
1001 100
0
--------
10001 10001
10001
-------
0

==Conversion to and from other numeral systems==

===Decimal to Binary===
[[File:Decimal to Binary Conversion.gif|alt=|frame|Conversion of (357)10 to binary notation results in (101100101)]]
To convert from a base-10 integer to its base-2 (binary) equivalent, the number is divided by two. The remainder is the least-significant bit. The quotient is again divided by two; its remainder becomes the next least significant bit. This process repeats until a quotient of one is reached. The sequence of remainders (including the final quotient of one) forms the binary value, as each remainder must be either zero or one when dividing by two. For example, (357)10 is expressed as (101100101)2.

=== Binary to Decimal ===
Conversion from base-2 to base-10 simply inverts the preceding algorithm. The bits of the binary number are used one by one, starting with the most significant (leftmost) bit. Beginning with the value 0, the prior value is doubled, and the next bit is then added to produce the next value. This can be organized in a multi-column table. For example, to convert 100101011012 to decimal:

{| style= "border: 1px solid #a2a9b1; border-spacing: 3px; background-color: #f8f9fa; color: black; margin: 0.5em 0 0.5em 1em; padding: 0.2em; line-height: 1.5em; width:22em"
!Prior value
! style="text-align:left" | × 2 +
!Next bit
!Next value
|-
|align="right"|0 ||× 2 +|| '''1''' || = 1
|-
|align="right"|1 ||× 2 +|| '''0''' || = 2
|-
|align="right"|2 ||× 2 +|| '''0''' || = 4
|-
|align="right"|4 ||× 2 +|| '''1''' || = 9
|-
|align="right"|9 ||× 2 +|| '''0''' || = 18
|-
|align="right"|18 ||× 2 +|| '''1''' || = 37
|-
|align="right"|37 ||× 2 +|| '''0''' || = 74
|-
|align="right"|74 ||× 2 +|| '''1''' || = 149
|-
|align="right"|149 ||× 2 +|| '''1''' || = 299
|-
|align="right"|299 ||× 2 +|| '''0''' || = 598
|-
|align="right"|598 ||× 2 +|| '''1''' || = '''1197'''
|}

The result is 119710. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the Horner scheme.

{|
! Binary 
| 1 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 || 0 || 1 ||
|-
! Decimal 
| 1×210 + || 0×29 + || 0×28 + || 1×27 + || 0×26 + || 1×25 + || 0×24 + || 1×23 + || 1×22 + || 0×21 + || 1×20 = || 1197
|}

The fractional parts of a number are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving.

In a fractional binary number such as 0.110101101012, the first digit is <math>\begin{matrix} \frac{1}{2} \end{matrix}</math>, the second <math>\begin{matrix} (\frac{1}{2})^2 = \frac{1}{4} \end{matrix}</math>, etc. So if there is a 1 in the first place after the decimal, then the number is at least <math>\begin{matrix} \frac{1}{2} \end{matrix}</math>, and vice versa. Double that number is at least 1. This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part.

For example, <math>\begin{matrix} (\frac{1}{3}) \end{matrix}</math>10, in binary, is:

{| class="wikitable"
!Converting!!Result
|-
|<math>\begin{matrix} \frac{1}{3} \end{matrix}</math> || 0.
|-
|<math>\begin{matrix} \frac{1}{3} \times 2 = \frac{2}{3} < 1 \end{matrix}</math> || 0.0
|-
|<math>\begin{matrix} \frac{2}{3} \times 2 = 1\frac{1}{3} \ge 1 \end{matrix}</math> || 0.01
|-
|<math>\begin{matrix} \frac{1}{3} \times 2 = \frac{2}{3} < 1 \end{matrix}</math> || 0.010
|-
|<math>\begin{matrix} \frac{2}{3} \times 2 = 1\frac{1}{3} \ge 1 \end{matrix}</math> || 0.0101
|}

Thus the repeating decimal fraction <math>0. \overline{3}</math>... is equivalent to the repeating binary fraction <math>0. \overline{01}</math>... .

Or for example, 0.110, in binary, is:

{| class="wikitable"
! Converting !! Result
|-
| '''0.1''' || 0.
|-
|0.1 × 2 = '''0.2''' < 1 || 0.0
|-
|0.2 × 2 = '''0.4''' < 1 || 0.00
|-
|0.4 × 2 = '''0.8''' < 1 || 0.000
|-
|0.8 × 2 = '''1.6''' ≥ 1 || 0.0001
|-
|0.6 × 2 = '''1.2''' ≥ 1 || 0.00011
|-
|0.2 × 2 = '''0.4''' < 1 || 0.000110
|-
|0.4 × 2 = '''0.8''' < 1 || 0.0001100
|-
|0.8 × 2 = '''1.6''' ≥ 1 || 0.00011001
|-
|0.6 × 2 = '''1.2''' ≥ 1 || 0.000110011
|-
|0.2 × 2 = '''0.4''' < 1 || 0.0001100110
|}

This is also a repeating binary fraction <math> 0.0 \overline{0011}</math>... . It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 0.1 + ... + 0.1, (10 additions) differs from 1 in floating point arithmetic. In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which 1/10 is not.

The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then divide by the appropriate power of two in the decimal base. For example:

: <math>
\begin{align}
x & = & 1100&.1\overline{01110}\ldots \\
x\times 2^6 & = & 1100101110&.\overline{01110}\ldots \\
x\times 2 & = & 11001&.\overline{01110}\ldots \\
x\times(2^6-2) & = & 1100010101 \\
x & = & 1100010101/111110 \\
x & = & (789/62)_{10}
\end{align}
</math>

Another way of converting from binary to decimal, often quicker for a person familiar with hexadecimal, is to do so indirectly—first converting (<math>x</math> in binary) into (<math>x</math> in hexadecimal) and then converting (<math>x</math> in hexadecimal) into (<math>x</math> in decimal).

For very large numbers, these simple methods are inefficient because they perform a large number of multiplications or divisions where one operand is very large. A simple divide-and-conquer algorithm is more effective asymptotically: given a binary number, it is divided by 10''k'', where ''k'' is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two are concatenated. Given a decimal number, it can be split into two pieces of about the same size, each of which is converted to binary, whereupon the first converted piece is multiplied by 10''k'' and added to the second converted piece, where ''k'' is the number of decimal digits in the second, least-significant piece before conversion.

===Hexadecimal===
Binary may be converted to and from hexadecimal more easily. This is because the radix of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 24, so it takes four digits of binary to represent one digit of hexadecimal, as shown in the adjacent table.

To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:

:3A16 = 0011 10102
:E716 = 1110 01112

To convert a binary number into its hexadecimal equivalent, divide it into groups of four bits. If the number of bits isn't a multiple of four, simply insert extra '''0''' bits at the left (called padding). For example:

:10100102 = 0101 0010 grouped with padding = 5216
:110111012 = 1101 1101 grouped = DD16

To convert a hexadecimal number into its decimal equivalent, multiply the decimal equivalent of each hexadecimal digit by the corresponding power of 16 and add the resulting values:

:C0E716 = (12 × 163) + (0 × 162) + (14 × 161) + (7 × 160) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,38310

===Octal===
Binary is also easily converted to the octal numeral system, since octal uses a radix of 8, which is a power of two (namely, 23, so it takes exactly three binary digits to represent an octal digit). The correspondence between octal and binary numerals is the same as for the first eight digits of hexadecimal in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth.

{| class="wikitable" style="text-align:center"
!Octal!!Binary
|-
| 0 || 000
|-
| 1 || 001
|-
| 2 || 010
|-
| 3 || 011
|-
| 4 || 100
|-
| 5 || 101
|-
| 6 || 110
|-
| 7 || 111
|}

Converting from octal to binary proceeds in the same fashion as it does for hexadecimal:

:658 = 110 1012
:178 = 001 1112

And from binary to octal:

:1011002 = 101 1002 grouped = 548
:100112 = 010 0112 grouped with padding = 238

And from octal to decimal:

:658 = (6 × 81) + (5 × 80) = (6 × 8) + (5 × 1) = 5310
:1278 = (1 × 82) + (2 × 81) + (7 × 80) = (1 × 64) + (2 × 8) + (7 × 1) = 8710

==Representing real numbers==

Non-integers can be represented by using negative powers, which are set off from the other digits by means of a radix point (called a decimal point in the decimal system). For example, the binary number 11.012 means:

{|
|'''1''' × 21 || (1 × 2 = '''2''') || plus
|-
|'''1''' × 20 || (1 × 1 = '''1''') || plus
|-
|'''0''' × 2−1 || (0 × <math> \tfrac{1}{2} </math> = '''0''') || plus
|-
|'''1''' × 2−2 || (1 × <math> \tfrac{1}{4} </math> = '''0.25''')
|}

For a total of 3.25 decimal.

All dyadic rational numbers <math>\frac{p}{2^a}</math> have a ''terminating'' binary numeral—the binary representation has a finite number of terms after the radix point. Other rational numbers have binary representation, but instead of terminating, they ''recur'', with a finite sequence of digits repeating indefinitely. For instance

:<math>\frac{1_{10}}{3_{10}} = \frac{1_2}{11_2} = 0.01010101\overline{01}\ldots\,_2 </math>

:<math>\frac{12_{10}}{17_{10}} = \frac{1100_2}{10001_2} = 0.10110100 10110100\overline{10110100}\ldots\,_2 </math>

The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in decimal. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the geometric series 2−1 + 2−2 + 2−3 + ... which is 1.

Binary numerals which neither terminate nor recur represent irrational numbers. For instance,
* 0.10100100010000100000100... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
* 1.0110101000001001111001100110011111110... is the binary representation of <math>\sqrt{2}</math>, the square root of 2, another irrational. It has no discernible pattern.

= Base 5 =
'''Quinary''' /ˈkwaɪnəri/ ('''base-5''' or '''pental''') is a numeral system with five as the base. A possible origination of a quinary system is that there are five digits on either hand.

In the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100 and sixty is written as 220.

As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two highly composite numbers (4 and 6) guarantees that many recurring fractions have relatively short periods.

Today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a sub-base system, is sexagesimal, base 60, which used 10 as a sub-base.

Each quinary digit can hold log25 (approx. 2.32) bits of information.

==Comparison to other radices==
{| class="wikitable" style="float:right; text-align:center"
|+ A quinary multiplication table
|-
| × || '''1''' || '''2''' || '''3''' || '''4''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''20'''
|-
| '''1''' || 1 || 2 || 3 || 4 || 10 || 11 || 12 || 13 || 14 || 20
|-
| '''2''' || 2 || 4 || 11 || 13 || 20 || 22 || 24 || 31 || 33 || 40
|-
| '''3''' || 3 || 11 || 14 || 22 || 30 || 33 || 41 || 44 || 102 || 110
|-
| '''4''' || 4 || 13 || 22 || 31 || 40 || 44 || 103 || 112 || 121 || 130
|-
| '''10''' || 10 || 20 || 30 || 40 || 100 || 110 || 120 || 130 || 140 || 200
|-
| '''11''' || 11 || 22 || 33 || 44 || 110 || 121 || 132 || 143 || 204 || 220
|-
| '''12''' || 12 || 24 || 41 || 103 || 120 || 132 || 144 || 211 || 223 || 240
|-
| '''13''' || 13 || 31 || 44 || 112 || 130 || 143 || 211 || 224 || 242 || 310
|-
| '''14''' || 14 || 33 || 102 || 121 || 140 || 204 || 223 || 242 || 311 || 330
|-
| '''20''' || 20 || 40 || 110 || 130 || 200 || 220 || 240 || 310 || 330 || 400
|}

{| class="wikitable"
|+ '''Numbers zero to twenty-five in standard quinary'''
|- align="center"
! Quinary
| 0 || 1 || 2 || 3 || 4 || 10 || 11 || 12 || 13 || 14 || 20 || 21 || 22
|- align="center"
! Binary
| 0 || 1 || 10 || 11 || 100 || 101 || 110 || 111 || 1000 || 1001 || 1010 || 1011 || 1100
|- align="center"
! Decimal
! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12
|- align="center"
!
|- align="center"
! Quinary
| 23 || 24 || 30 || 31 || 32 || 33 || 34 || 40 || 41 || 42 || 43 || 44 || 100
|- align="center"
! Binary
| 1101 || 1110 || 1111 || 10000 || 10001 || 10010 || 10011 || 10100 || 10101 || 10110 || 10111 || 11000 || 11001
|- align="center"
! Decimal
! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20 !! 21 !! 22 !! 23 !! 24 !! 25
|}

{| class="wikitable"
|+ '''Fractions in quinary'''
|'''Decimal''' (periodic part) ||'''Quinary''' (periodic part)
|'''Binary''' (periodic part)
|-
|1/2 = 0.5
|'''1/2''' = 0.2
|1/10 = 0.1
|-
|1/3 = 0.3
|'''1/3''' = 0.13
|1/11 = 0.01
|-
|1/4 = 0.25
|'''1/4''' = 0.1
|1/100 = 0.01
|-
|1/5 = 0.2
|'''1/10''' = 0.1
|1/101 = 0.0011
|-
|1/6 = 0.16
|'''1/11''' = 0.04
|1/110 = 0.010
|-
|1/7 = 0.142857
|'''1/12''' = 0.032412
|1/111 = 0.001
|-
|1/8 = 0.125
|'''1/13''' = 0.03
|1/1000 = 0.001
|-
|1/9 = 0.1
|'''1/14''' = 0.023421
|1/1001 = 0.000111
|-
|1/10 = 0.1
|'''1/20''' = 0.02
|1/1010 = 0.00011
|-
|1/11 = 0.09
|'''1/21''' = 0.02114
|1/1011 = 0.0001011101
|-
|1/12 = 0.083
|'''1/22''' = 0.02
|1/1100 = 0.0001
|-
|1/13 = 0.076923
|'''1/23''' = 0.0143
|1/1101 = 0.000100111011
|-
|1/14 = 0.0714285
|'''1/24''' = 0.013431
|1/1110 = 0.0001
|-
|1/15 = 0.06
|'''1/30''' = 0.013
|1/1111 = 0.0001
|-
|1/16 = 0.0625
|'''1/31''' = 0.0124
|1/10000 = 0.0001
|-
|1/17 = 0.0588235294117647
|'''1/32''' = 0.0121340243231042
|1/10001 = 0.00001111
|-
|1/18 = 0.05
|'''1/33''' = 0.011433
|1/10010 = 0.0000111
|-
|1/19 = 0.052631578947368421
|'''1/34''' = 0.011242141
|1/10011 = 0.000011010111100101
|-
|1/20 = 0.05
|'''1/40''' = 0.01
|1/10100 = 0.000011
|-
|1/21 = 0.047619
|'''1/41''' = 0.010434
|1/10101 = 0.000011
|-
|1/22 = 0.045
|'''1/42''' = 0.01032
|1/10110 = 0.00001011101
|-
|1/23 = 0.0434782608695652173913
|'''1/43''' = 0.0102041332143424031123
|1/10111 = 0.00001011001
|-
|1/24 = 0.0416
|'''1/44''' = 0.01
|1/11000 = 0.00001
|-
|1/25 = 0.04
|'''1/100''' = 0.01
|1/11001 = 0.00001010001111010111
|}

==Usage==
Many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below:

{| class="wikitable" border="1" style="text-align:center"
! Number !! Base 5 !! Numeral
|-
! style="text-align:right" | 1
| 1
| wanggany
|-
! style="text-align:right" | 2
| 2
| marrma
|-
! style="text-align:right" | 3
| 3
| lurrkun
|-
! style="text-align:right" | 4
| 4
| dambumiriw
|-
! style="text-align:right" | 5
| 10
| wanggany rulu
|-
! style="text-align:right" | 10
| 20
| marrma rulu
|-
! style="text-align:right" | 15
| 30
| lurrkun rulu
|-
! style="text-align:right" | 20
| 40
| dambumiriw rulu
|-
! style="text-align:right" | 25
| 100
| dambumirri rulu
|-
! style="text-align:right" | 50
| 200
| marrma dambumirri rulu
|-
! style="text-align:right" | 75
| 300
| lurrkun dambumirri rulu
|-
! style="text-align:right" | 100
| 400
| dambumiriw dambumirri rulu
|-
! style="text-align:right" | 125
| 1000
| dambumirri dambumirri rulu
|-
! style="text-align:right" | 625
| 10000

| dambumirri dambumirri dambumirri rulu
|}

==Biquinary==
[[File:Chinese-abacus.jpg|thumb|right|Chinese Abacus or suanpan]]
A decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are an early biquinary system. The numbers 1, 5, 10, and 50 are written as '''I''', '''V''', '''X''', and '''L''' respectively. Seven is '''VII''' and seventy is '''LXX'''. The full list is:
{| class="wikitable" style="text-align:center"
| '''I''' || '''V''' || '''X''' || '''L''' || '''C''' || '''D''' || '''M'''
|-
| 1 || 5 || 10 || 50 || 100 || 500 || 1000
|}
Note that these are not positional number systems. In theory a number such as 73 could be written as IIIXXL without ambiguity as well as LXXIII and it is still not possible to extend it beyond thousands. There is also no sign for zero. But with the introduction of inversions such as IV and IX, it was necessary to keep the order from most to least significant.

Many versions of the abacus, such as the suanpan and soroban, use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary. Units of currencies are commonly partially or wholly biquinary.

Bi-quinary coded decimal is a variant of biquinary that was used on a number of early computers including Colossus and the IBM 650 to represent decimal numbers.

= Base 10 =

The '''decimal''' numeral system (also called the '''base-ten''' positional numeral system, and occasionally called '''denary''' /ˈdiːnəri/ or '''decanary''') is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''.

A ''decimal numeral'' (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in 25.9703 or 3,1415). ''Decimal'' may also refer specifically to the digits after the decimal separator, such as in "3.14 is the approximation of <math> \pi </math> to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.

The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form {{math|''a''/10''n''}}, where {{math|''a''}} is an integer, and {{math|''n''}} is a non-negative integer.

The decimal system has been extended to ''infinite decimals'' for representing any real number, by using an infinite sequence of digits after the decimal separator (see decimal representation). In this context, the decimal numerals with a finite number of non-zero digits after the decimal separator are sometimes called ''terminating decimals''. A ''repeating decimal'' is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., 5.123144144144144... = <math> 5.123 \overline{144}</math>). An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

==Origin==
[[File:Two hand, ten fingers.jpg|thumb|right|Ten fingers on two hands, the possible origin of decimal counting]]
Many numeral systems of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the Egyptian numerals, then the Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called ''decimal fractions'' or ''decimal numbers'', for forming the ''decimal numeral system''.

== Decimal notation ==
For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "−". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; and a comma "{{math|,}}" in other countries.

For representing a non-negative number, a decimal numeral consists of
* either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer,
*:<math>a_ma_{m-1}\ldots a_0</math>
*or a decimal mark separating two sequences of digits (such as "20.70828")
::<math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math>.
If {{math|''m'' > 0}}, that is, if the first sequence contains at least two digits, it is generally assumed that the first digit {{math|''a''''m''}} is not zero. In some circumstances it may be useful to have one or more 0's on the left; this does not change the value represented by the decimal: for example, {{math|1=3.14 = 03.14 = 003.14}}. Similarly, if the final digit on the right of the decimal mark is zero—that is, if {{math|1=''b''''n'' = 0}}—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; for example, {{math|1=15 = 15.0 = 15.00}} and {{math|1=5.2 = 5.20 = 5.200}}.

For representing a negative number, a minus sign is placed before {{math|''a''''m''}}.

The numeral <math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math> represents the number
:<math>a_m10^m+a_{m-1}10^{m-1}+\cdots+a_{0}10^0+\frac{b_1}{10^1}+\frac{b_2}{10^2}+\cdots+\frac{b_n}{10^n}</math>.
The ''integer part'' or ''integral part'' of a decimal numeral is the integer written to the left of the decimal separator (see also truncation). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the ''fractional part'', which equals the difference between the numeral and its integer part.

When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example {{math|.1234}}, instead of {{math|0.1234}}). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation.

In brief, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is a positional numeral system.

== Decimal fractions ==
'''Decimal fractions''' (sometimes called '''decimal numbers''', especially in contexts involving explicit fractions) are the rational numbers that may be expressed as a fraction whose denominator is a power of ten. For example, the decimals <math>0.8, 14.89, 0.00024, 1.618, 3.14159</math> represent the fractions <math> \tfrac{8}{10}</math>, <math> \tfrac{1489}{100}</math>, <math> \tfrac{24}{100000}</math>, <math>\tfrac {1618}{1000}</math> and <math>\tfrac{314159}{100000}</math>, and are therefore decimal numbers.

More generally, a decimal with {{math|''n''}} digits after the separator represents the fraction with denominator {{math|10''n''}}, whose numerator is the integer obtained by removing the separator.

It follows that a number is a decimal fraction if and only if it has a finite decimal representation.

Expressed as a fully reduced fraction, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are
:<math>1=2^0\cdot 5^0, 2=2^1\cdot 5^0, 4=2^2\cdot 5^0, 5=2^0\cdot 5^1, 8=2^3\cdot 5^0, 10=2^1\cdot 5^1, 16=2^4\cdot 5^0, 25=2^0\cdot 5^2, \ldots</math>

==Real number approximation==

Decimal numerals do not allow an exact representation for all real numbers, e.g. for the real number <math>\pi</math>. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates the real <math>\pi</math>, being less than 10−5 off; so decimals are widely used in science, engineering and everyday life.

More precisely, for every real number {{Mvar|x}} and every positive integer {{Mvar|n}}, there are two decimals {{Mvar|''L''}} and {{Mvar|''u''}} with at most ''{{Mvar|n}}'' digits after the decimal mark such that {{Math|''L'' ≤ ''x'' ≤ ''u''}} and {{Math|1=(''u'' − ''L'') = 10−''n''}}.

Numbers are very often obtained as the result of measurement. As measurements are subject to measurement uncertainty with a known upper bound, the result of a measurement is well-represented by a decimal with {{math|''n''}} digits after the decimal mark, as soon as the absolute measurement error is bounded from above by {{Math|10−''n''}}. In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also significant figures).

==Infinite decimal expansion==

For a real number {{Mvar|x}} and an integer {{Math|''n'' ≥ 0}}, let {{Math|[''x'']''n''}} denote the (finite) decimal expansion of the greatest number that is not greater than ''{{Mvar|x}}'' that has exactly {{Mvar|n}} digits after the decimal mark. Let {{Math|''d''''i''}} denote the last digit of {{Math|[''x'']''i''}}. It is straightforward to see that {{Math|[''x'']''n''}} may be obtained by appending {{Math|''d''''n''}} to the right of {{Math|[''x'']''n''−1}}. This way one has
:{{Math|1=[''x'']''n'' = [''x'']0.''d''1''d''2...''d''''n''−1''d''''n''}},

and the difference of {{Math|[''x'']''n''−1}} and {{Math|[''x'']''n''}} amounts to
:<math>\left\vert \left [ x \right ]_n-\left [ x \right ]_{n-1} \right\vert=d_n\cdot10^{-n}<10^{-n+1}</math>,

which is either 0, if {{Math|1=''d''''n'' = 0}}, or gets arbitrarily small as ''{{Mvar|n}}'' tends to infinity. According to the definition of a limit, ''{{Mvar|x}}'' is the limit of {{Math|[''x'']''n''}} when ''{{Mvar|n}}'' tends to infinity. This is written as<math display="inline">\; x = \lim_{n\rightarrow\infty} [x]_n \;</math>or
: {{Math|1=''x'' = [''x'']0.''d''1''d''2...''d''''n''...}},
which is called an '''infinite decimal expansion''' of ''{{Mvar|x}}''.

Conversely, for any integer {{Math|[''x'']0}} and any sequence of digits<math display="inline">\;(d_n)_{n=1}^{\infty}</math> the (infinite) expression {{Math|[''x'']0.''d''1''d''2...''d''''n''...}} is an ''infinite decimal expansion'' of a real number ''{{Mvar|x}}''. This expansion is unique if neither all {{Math|''d''''n''}} are equal to 9 nor all {{Math|''d''''n''}} are equal to 0 for ''{{Mvar|n}}'' large enough (for all ''{{Mvar|n}}'' greater than some natural number {{Mvar|N}}).

If all {{Math|''d''''n''}} for {{Math|''n'' > ''N''}} equal to 9 and {{Math|1=[''x'']''n'' = [''x'']0.''d''1''d''2...''d''''n''}}, the limit of the sequence<math display="inline">\;([x]_n)_{n=1}^{\infty}</math> is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: {{Math|''d''''N''}}, by {{Math|''d''''N'' + 1}}, and replacing all subsequent 9s by 0s (see 0.999...).

Any such decimal fraction, i.e.: {{Math|1=''d''''n'' = 0}} for {{Math|''n'' > ''N''}}, may be converted to its equivalent infinite decimal expansion by replacing {{Math|''d''''N''}} by {{Math|''d''''N'' − 1}} and replacing all subsequent 0s by 9s (see 0.999...).

In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of {{Math|[''x'']''n''}}, and the other containing only 9s after some place, which is obtained by defining {{Math|[''x'']''n''}} as the greatest number that is ''less'' than {{Mvar|x}}, having exactly ''{{Mvar|n}}'' digits after the decimal mark.

=== Rational numbers ===

Long division allows computing the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a ''repeating decimal''. For example,
:<math> \tfrac{1}{81}</math> = 0. 012345679 012... (with the group 012345679 indefinitely repeating).

The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational.
{|
|-
|For example, if ''x'' is || 0.4156156156...
|-
|then 10,000''x'' is || 4156.156156156...
|-
|and 10''x'' is|| 4.156156156...
|-
|so 10,000''x'' − 10''x'', i.e. 9,990''x'', is|| 4152.000000000...
|-
|and ''x'' is|| <math> \tfrac{4152}{9990}</math>
|}
or, dividing both numerator and denominator by 6, <math> \tfrac{692}{1665}</math>.

== Decimal computation ==

[[File:Decimal multiplication table.JPG|thumb|right|300px|Diagram of the world's earliest known multiplication table (c. 305 BCE) from the Warring States period]]

Most modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as the ENIAC or the IBM 650, used decimal representation internally).
For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems.

For most purposes, however, binary values are converted to or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)

Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of binary-coded decimal, especially in database implementations, but there are other decimal representations in use (including decimal floating point such as in newer revisions of the IEEE 754|IEEE 754 Standard for Floating-Point Arithmetic).

Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of <math>10</math> have no finite binary fractional representation; and is generally impossible for multiplication (or division). See Arbitrary-precision arithmetic for exact calculations.

= Licensing =
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Radix Radix, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Binary_number Binary number, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Quinary Quinary, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Decimal Decimal, Wikipedia] under a CC BY-SA license

Base 10, Base 2 & Base 5

2022-02-05T19:03:04Z

Khanh: /* Base 2 */

In a positional numeral system, the '''radix''' or '''base''' is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9.

In any standard positional numeral system, a number is conventionally written as (''x'')''y'' with ''x'' as the string of digits and ''y'' as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)10 is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.

= In numeral systems =
In the system with radix 13, for example, a string of digits such as 398 denotes the (decimal) number 3 × 132 + 9 × 131 + 8 × 130 = 632.

More generally, in a system with radix ''b'' (''b'' > 1), a string of digits ''d''1 … ''dn'' denotes the number ''d''1''b''''n''−1 + ''d''2''b''''n''−2 + … + ''dnb''0, where 0 ≤ ''di'' < ''b''. In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radix ''b'' would have a ones' place, then a ''b''1s' place, a ''b''2s' place, etc.

Commonly used numeral systems include:
{| class="wikitable sortable"
! Base/radix
! Name
! Description
|-
| 2
| Binary numeral system
| Used internally by nearly all computers, is base 2. The two digits are "0" and "1", expressed from switches displaying OFF and ON, respectively. Used in most electric counters.
|-
| 8
| Octal system
| Used occasionally in computing. The eight digits are "0"–"7" and represent 3 bits (23).
|-
| 10
| Decimal system
| Used by humans in the vast majority of cultures. Its ten digits are "0"–"9". Used in most mechanical counters.
|-
| 12
| Duodecimal (dozenal) system
| Sometimes advocated due to divisibility by 2, 3, 4, and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
|-
| 16
| Hexadecimal system
| Often used in computing as a more compact representation of binary (1 hex digit per 4 bits). The sixteen digits are "0"–"9" followed by "A"–"F" or "a"–"f".
|-
| 20
| Vigesimal system
| Traditional numeral system in several cultures, still used by some for counting. Historically also known as the ''score system'' in English, now most famous in the phrase "four score and seven years ago" in the Gettysburg Address.
|-
| 60
| Sexagesimal system
| Originated in ancient Sumer and passed to the Babylonians. Used today as the basis of modern circular coordinate system (degrees, minutes, and seconds) and time measuring (minutes, and seconds) by analogy to the rotation of the Earth.
|}

= Base 2 =
A '''binary number''' is a number expressed in the '''base-2 numeral system''' or '''binary numeral system''', a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).

The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language.

==Representation==
Any number can be represented by a sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. Any of the following rows of symbols can be interpreted as the binary numeric value of 667:

{| style="text-align:center;"
| 1 || 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1 || 1
|-
| <nowiki>|</nowiki> || ― || <nowiki>|</nowiki> || ― || ― || <nowiki>|</nowiki> || <nowiki>|</nowiki> || ― || <nowiki>|</nowiki> || <nowiki>|</nowiki>
|-
| ☒ || ☐ || ☒ || ☐ || ☐ || ☒ || ☒ || ☐ || ☒ || ☒
|-
| y || n || y || n || n || y || y || n || y || y
|}

[[Image:Binary clock.svg|250px|thumbnail|right|A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.]]

The numeric value represented in each case is dependent upon the value assigned to each symbol. In the earlier days of computing, switches, punched holes and punched paper tapes were used to represent binary values. In a modern computer, the numeric values may be represented by two different voltages; on a magnetic disk, magnetic polarities may be used. A "positive", "yes", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.

In keeping with customary representation of numerals using Arabic numerals, binary numbers are commonly written using the symbols '''0''' and '''1'''. When written, binary numerals are often subscripted, prefixed or suffixed in order to indicate their base, or radix. The following notations are equivalent:
* 100101 binary (explicit statement of format)
* 100101b (a suffix indicating binary format; also known as Intel convention)
* 100101B (a suffix indicating binary format)
* bin 100101 (a prefix indicating binary format)
* 1001012 (a subscript indicating base-2 (binary) notation)
* %100101 (a prefix indicating binary format; also known as Motorola convention)
* 0b100101 (a prefix indicating binary format, common in programming languages)
* 6b100101 (a prefix indicating number of bits in binary format, common in programming languages)
* #b100101 (a prefix indicating binary format, common in Lisp programming languages)

When spoken, binary numerals are usually read digit-by-digit, in order to distinguish them from decimal numerals. For example, the binary numeral 100 is pronounced ''one zero zero'', rather than ''one hundred'', to make its binary nature explicit, and for purposes of correctness. Since the binary numeral 100 represents the value four, it would be confusing to refer to the numeral as ''one hundred'' (a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct ''value''), but this does not make its binary nature explicit.

==Counting in binary==
Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar decimal counting system as a frame of reference.

===Decimal counting===
Decimal counting uses the ten symbols ''0'' through ''9''. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the ''first digit''. When the available symbols for this position are exhausted, the least significant digit is reset to ''0'', and the next digit of higher significance (one position to the left) is incremented (''overflow''), and incremental substitution of the low-order digit resumes. This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows:

:000, 001, 002, ... 007, 008, 009, (rightmost digit is reset to zero, and the digit to its left is incremented)
:0'''1'''0, 011, 012, ...
:   ...
:090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented)
:'''1'''00, 101, 102, ...

===Binary counting===
[[File:Binary counter.gif|thumb|This counter shows how to count in binary from numbers zero through thirty-one.]]
[[File:Binary_guess_number_trick_SMIL.svg|thumb|upright|link={{filepath:binary_guess_number_trick_SMIL.svg}}|A party trick to guess a number from which cards it is printed on uses the bits of the binary representation of the number. In the SVG file, click a card to toggle it]]
Binary counting follows the same procedure, except that only the two symbols ''0'' and ''1'' are available. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:

:0000,
:000'''1''', (rightmost digit starts over, and next digit is incremented)
:00'''1'''0, 0011, (rightmost two digits start over, and next digit is incremented)
:0'''1'''00, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented)
:'''1'''000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...

In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" digit. For example, the binary number 100101 is converted to decimal form as follows:

:1001012 = [ ( '''1''' ) × 25 ] + [ ( '''0''' ) × 24 ] + [ ( '''0''' ) × 23 ] + [ ( '''1''' ) × 22 ] + [ ( '''0''' ) × 21 ] + [ ( '''1''' ) × 20 ]

:1001012 = [ '''1''' × 32 ] + [ '''0''' × 16 ] + [ '''0''' × 8 ] + [ '''1''' × 4 ] + [ '''0''' × 2 ] + [ '''1''' × 1 ]

:'''1001012 = 3710'''

==Fractions==

Fractions in binary arithmetic terminate only if 2 is the only prime factor in the denominator. As a result, 1/10 does not have a finite binary representation ('''10''' has prime factors '''2''' and '''5'''). This causes 10 × 0.1 not to precisely equal 1 in floating-point arithmetic. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × '''2−1''' + 1 × '''2−2''' + 0 × '''2−3''' + 1 × '''2−4''' + ... = 0.3125 + ... An exact value cannot be found with a sum of a finite number of inverse powers of two, the zeros and ones in the binary representation of 1/3 alternate forever.

{| class="wikitable"
|-
! Fraction
! [[Base 10|Decimal]]
! Binary
! Fractional approximation
|-
| 1/1
| 1{{pad|0.25em}}or{{pad|0.25em}}0.999...
| 1{{pad|0.25em}}or{{pad|0.25em}}0.111...
| 1/2 + 1/4 + 1/8...
|-
| 1/2
| 0.5{{pad|0.25em}}or{{pad|0.25em}}0.4999...
| 0.1{{pad|0.25em}}or{{pad|0.25em}}0.0111...
| 1/4 + 1/8 + 1/16 . . .
|-
| 1/3
| 0.333...
| 0.010101...
| 1/4 + 1/16 + 1/64 . . .
|-
| 1/4
| 0.25{{pad|0.25em}}or{{pad|0.25em}}0.24999...
| 0.01{{pad|0.25em}}or{{pad|0.25em}}0.00111...
| 1/8 + 1/16 + 1/32 . . .
|-
| 1/5
| 0.2{{pad|0.25em}}or{{pad|0.25em}}0.1999...
| 0.00110011...
| 1/8 + 1/16 + 1/128 . . .
|-
| 1/6
| 0.1666...
| 0.0010101...
| 1/8 + 1/32 + 1/128 . . .
|-
| 1/7
| 0.142857142857...
| 0.001001...
| 1/8 + 1/64 + 1/512 . . .
|-
| 1/8
| 0.125{{pad|0.25em}}or{{pad|0.25em}}0.124999...
| 0.001{{pad|0.25em}}or{{pad|0.25em}}0.000111...
| 1/16 + 1/32 + 1/64 . . .
|-
| 1/9
| 0.111...
| 0.000111000111...
| 1/16 + 1/32 + 1/64 . . .
|-
| 1/10
| 0.1{{pad|0.25em}}or{{pad|0.25em}}0.0999...
| 0.000110011...
| 1/16 + 1/32 + 1/256 . . .
|-
| 1/11
| 0.090909...
| 0.00010111010001011101...
| 1/16 + 1/64 + 1/128 . . .
|-
| 1/12
| 0.08333...
| 0.00010101...
| 1/16 + 1/64 + 1/256 . . .
|-
| 1/13
| 0.076923076923...
| 0.000100111011000100111011...
| 1/16 + 1/128 + 1/256 . . .
|-
| 1/14
| 0.0714285714285...
| 0.0001001001...
| 1/16 + 1/128 + 1/1024 . . .
|-
| 1/15
| 0.0666...
| 0.00010001...
| 1/16 + 1/256 . . .
|-
| 1/16
| 0.0625{{pad|0.25em}}or{{pad|0.25em}}0.0624999...
| 0.0001{{pad|0.25em}}or{{pad|0.25em}}0.0000111...
| 1/32 + 1/64 + 1/128 . . .
|}

==Binary arithmetic==
Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.

===Addition===
[[Image:Half Adder.svg|thumbnail|200px|right|The circuit diagram for a binary half adder, which adds two bits together, producing sum and carry bits]]

The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:

:0 + 0 → 0
:0 + 1 → 1
:1 + 0 → 1
:1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21) )
Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:

:5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101) )
:7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101) )

This is known as ''carrying''. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:

1 1 1 1 1 (carried digits)
0 1 1 0 1
+ 1 0 1 1 1
-------------
= 1 0 0 1 0 0 = 36

In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 102. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 102 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 112. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).

When computers must add two numbers, the rule that:
x xor y = (x + y) mod 2
for any two bits x and y allows for very fast calculation, as well.

====Long carry method====
A simplification for many binary addition problems is the Long Carry Method or Brookhouse Method of Binary Addition. This method is generally useful in any binary addition in which one of the numbers contains a long "string" of ones. It is based on the simple premise that under the binary system, when given a "string" of digits composed entirely of ''n'' ones (where ''n'' is any integer length), adding 1 will result in the number 1 followed by a string of ''n'' zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of ''n'' 9s will result in the number 1 followed by a string of ''n'' 0s:

Binary Decimal
1 1 1 1 1 likewise 9 9 9 9 9
+ 1 + 1
——————————— ———————————
1 0 0 0 0 0 1 0 0 0 0 0

Such long strings are quite common in the binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations. In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 02 (95810) and 1 0 1 0 1 1 0 0 1 12 (69110), using the traditional carry method on the left, and the long carry method on the right:

Traditional Carry Method Long Carry Method
vs.
1 1 1 1 1 1 1 1 (carried digits) 1 ← 1 ← carry the 1 until it is one digit past the "string" below
1 1 1 0 1 1 1 1 1 0 <s>1 1 1</s> 0 <s>1 1 1 1 1</s> 0 cross out the "string",
+ 1 0 1 0 1 1 0 0 1 1 + 1 0 <s>1</s> 0 1 1 0 0 <s>1</s> 1 and cross out the digit that was added to it
——————————————————————— ——————————————————————
= 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1

The top row shows the carry bits used. Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series. The "used" numbers must be crossed off, since they are already added. Other long strings may likewise be cancelled using the same technique. Then, simply add together any remaining digits normally. Proceeding in this manner gives the final answer of 1 1 0 0 1 1 1 0 0 0 12 (164910). In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort.

====Addition table====

{| class="wikitable" style="text-align:center"
|-
! style="width:1.5em" |
! style="width:1.5em" | 0
! style="width:1.5em" | 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 10
|}

The binary addition table is similar, but not the same, as the truth table of the logical disjunction operation <math>\lor</math>. The difference is that <math>1</math><math>\lor </math><math>1=1</math>, while <math>1+1=10</math>.

=== Subtraction ===

Subtraction works in much the same way:

:0 − 0 → 0
:0 − 1 → 1, borrow 1
:1 − 0 → 1
:1 − 1 → 0
Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as ''borrowing''. The principle is the same as for carrying. When the result of a subtraction is less than 0, the least possible value of a digit, the procedure is to "borrow" the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value.

* * * * (starred columns are borrowed from)
1 1 0 1 1 1 0
− 1 0 1 1 1
----------------
= 1 0 1 0 1 1 1

* (starred columns are borrowed from)
1 0 1 1 1 1 1
- 1 0 1 0 1 1
----------------
= 0 1 1 0 1 0 0

Subtracting a positive number is equivalent to ''adding'' a negative number of equal absolute value. Computers use signed number representations to handle negative numbers—most commonly the two's complement notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula:

: {{math|1=A − B = A + not B + 1}}

===Multiplication===
Multiplication in binary is similar to its decimal counterpart. Two numbers ''A'' and ''B'' can be multiplied by partial products: for each digit in ''B'', the product of that digit in ''A'' is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in ''B'' that was used. The sum of all these partial products gives the final result.

Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:
* If the digit in ''B'' is 0, the partial product is also 0
* If the digit in ''B'' is 1, the partial product is equal to ''A''

For example, the binary numbers 1011 and 1010 are multiplied as follows:

1 0 1 1 (''A'')
× 1 0 1 0 (''B'')
---------
0 0 0 0 ← Corresponds to the rightmost 'zero' in ''B''
+ 1 0 1 1 ← Corresponds to the next 'one' in ''B''
+ 0 0 0 0
+ 1 0 1 1
---------------
= 1 1 0 1 1 1 0

Binary numbers can also be multiplied with bits after a binary point:

1 0 1 . 1 0 1 ''A'' (5.625 in decimal)
× 1 1 0 . 0 1 ''B'' (6.25 in decimal)
-------------------
1 . 0 1 1 0 1 ← Corresponds to a 'one' in ''B''
+ 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in ''B''
+ 0 0 0 . 0 0 0
+ 1 0 1 1 . 0 1
+ 1 0 1 1 0 . 1
---------------------------
= 1 0 0 0 1 1 . 0 0 1 0 1 (35.15625 in decimal)

See also Booth's multiplication algorithm.

====Multiplication table====
{| class="wikitable" style="text-align:center"
|-
! style="width:1.5em" |
! style="width:1.5em" | 0
! style="width:1.5em" | 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

The binary multiplication table is the same as the truth table of the logical conjunction operation <math>\land</math>.

===Division===

Long division in binary is again similar to its decimal counterpart.

In the example below, the divisor is 1012, or 5 in decimal, while the dividend is 110112, or 27 in decimal. The procedure is the same as that of decimal long division; here, the divisor 1012 goes into the first three digits 1102 of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a "1") is included to obtain a new three-digit sequence:

1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
0 0 1

The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:

1 0 1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
1 1 1
− 1 0 1
-----
0 1 0

Thus, the quotient of 110112 divided by 1012 is 1012, as shown on the top line, while the remainder, shown on the bottom line, is 102. In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2.

Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result.

===Square root===
The process of taking a binary square root digit by digit is the same as for a decimal square root and is explained here. An example is:

1 0 0 1
---------
√ 1010001
1
---------
101 01
0
--------
1001 100
0
--------
10001 10001
10001
-------
0

==Conversion to and from other numeral systems==

===Decimal to Binary===
[[File:Decimal to Binary Conversion.gif|alt=|frame|Conversion of (357)10 to binary notation results in (101100101)]]
To convert from a base-10 [[Integer (computer science)|integer]] to its base-2 (binary) equivalent, the number is [[division by two|divided by two]]. The remainder is the [[least-significant bit]]. The quotient is again divided by two; its remainder becomes the next least significant bit. This process repeats until a quotient of one is reached. The sequence of remainders (including the final quotient of one) forms the binary value, as each remainder must be either zero or one when dividing by two. For example, (357)10 is expressed as (101100101)2.<ref>{{Cite web|url=https://www.chalkstreet.com/aptipedia/knowledgebase/base-system/|title=Base System|access-date=31 August 2016}}</ref>

=== Binary to Decimal ===
Conversion from base-2 to base-10 simply inverts the preceding algorithm. The bits of the binary number are used one by one, starting with the most significant (leftmost) bit. Beginning with the value 0, the prior value is doubled, and the next bit is then added to produce the next value. This can be organized in a multi-column table. For example, to convert 100101011012 to decimal:

{| style= "border: 1px solid #a2a9b1; border-spacing: 3px; background-color: #f8f9fa; color: black; margin: 0.5em 0 0.5em 1em; padding: 0.2em; line-height: 1.5em; width:22em"
!Prior value
! style="text-align:left" | × 2 +
!Next bit
!Next value
|-
|align="right"|0 ||× 2 +|| '''1''' || = 1
|-
|align="right"|1 ||× 2 +|| '''0''' || = 2
|-
|align="right"|2 ||× 2 +|| '''0''' || = 4
|-
|align="right"|4 ||× 2 +|| '''1''' || = 9
|-
|align="right"|9 ||× 2 +|| '''0''' || = 18
|-
|align="right"|18 ||× 2 +|| '''1''' || = 37
|-
|align="right"|37 ||× 2 +|| '''0''' || = 74
|-
|align="right"|74 ||× 2 +|| '''1''' || = 149
|-
|align="right"|149 ||× 2 +|| '''1''' || = 299
|-
|align="right"|299 ||× 2 +|| '''0''' || = 598
|-
|align="right"|598 ||× 2 +|| '''1''' || = '''1197'''
|}

The result is 119710. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the [[Horner scheme]].

{|
! Binary 
| 1 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 || 0 || 1 ||
|-
! Decimal 
| 1×210 + || 0×29 + || 0×28 + || 1×27 + || 0×26 + || 1×25 + || 0×24 + || 1×23 + || 1×22 + || 0×21 + || 1×20 = || 1197
|}

The fractional parts of a number are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving.

In a fractional binary number such as 0.110101101012, the first digit is <math>\begin{matrix} \frac{1}{2} \end{matrix}</math>, the second <math>\begin{matrix} (\frac{1}{2})^2 = \frac{1}{4} \end{matrix}</math>, etc. So if there is a 1 in the first place after the decimal, then the number is at least <math>\begin{matrix} \frac{1}{2} \end{matrix}</math>, and vice versa. Double that number is at least 1. This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part.

For example, <math>\begin{matrix} (\frac{1}{3}) \end{matrix}</math>10, in binary, is:

{| class="wikitable"
!Converting!!Result
|-
|<math>\begin{matrix} \frac{1}{3} \end{matrix}</math> || 0.
|-
|<math>\begin{matrix} \frac{1}{3} \times 2 = \frac{2}{3} < 1 \end{matrix}</math> || 0.0
|-
|<math>\begin{matrix} \frac{2}{3} \times 2 = 1\frac{1}{3} \ge 1 \end{matrix}</math> || 0.01
|-
|<math>\begin{matrix} \frac{1}{3} \times 2 = \frac{2}{3} < 1 \end{matrix}</math> || 0.010
|-
|<math>\begin{matrix} \frac{2}{3} \times 2 = 1\frac{1}{3} \ge 1 \end{matrix}</math> || 0.0101
|}

Thus the repeating decimal fraction 0.{{overline|3}}... is equivalent to the repeating binary fraction 0.{{overline|01}}... .

Or for example, 0.110, in binary, is:

{| class="wikitable"
! Converting !! Result
|-
| '''0.1''' || 0.
|-
|0.1 × 2 = '''0.2''' < 1 || 0.0
|-
|0.2 × 2 = '''0.4''' < 1 || 0.00
|-
|0.4 × 2 = '''0.8''' < 1 || 0.000
|-
|0.8 × 2 = '''1.6''' ≥ 1 || 0.0001
|-
|0.6 × 2 = '''1.2''' ≥ 1 || 0.00011
|-
|0.2 × 2 = '''0.4''' < 1 || 0.000110
|-
|0.4 × 2 = '''0.8''' < 1 || 0.0001100
|-
|0.8 × 2 = '''1.6''' ≥ 1 || 0.00011001
|-
|0.6 × 2 = '''1.2''' ≥ 1 || 0.000110011
|-
|0.2 × 2 = '''0.4''' < 1 || 0.0001100110
|}

This is also a repeating binary fraction 0.0{{overline|0011}}... . It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 0.1 + ... + 0.1, (10 additions) differs from 1 in [[floating point arithmetic]]. In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which 1/10 is not.

The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then divide by the appropriate power of two in the decimal base. For example:

: <math>
\begin{align}
x & = & 1100&.1\overline{01110}\ldots \\
x\times 2^6 & = & 1100101110&.\overline{01110}\ldots \\
x\times 2 & = & 11001&.\overline{01110}\ldots \\
x\times(2^6-2) & = & 1100010101 \\
x & = & 1100010101/111110 \\
x & = & (789/62)_{10}
\end{align}
</math>

Another way of converting from binary to decimal, often quicker for a person familiar with [[hexadecimal]], is to do so indirectly—first converting (<math>x</math> in binary) into (<math>x</math> in hexadecimal) and then converting (<math>x</math> in hexadecimal) into (<math>x</math> in decimal).

For very large numbers, these simple methods are inefficient because they perform a large number of multiplications or divisions where one operand is very large. A simple divide-and-conquer algorithm is more effective asymptotically: given a binary number, it is divided by 10''k'', where ''k'' is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two are [[Concatenation|concatenated]]. Given a decimal number, it can be split into two pieces of about the same size, each of which is converted to binary, whereupon the first converted piece is multiplied by 10''k'' and added to the second converted piece, where ''k'' is the number of decimal digits in the second, least-significant piece before conversion.

===Hexadecimal===
{{Main|Hexadecimal}}
{{Hexadecimal table}}
Binary may be converted to and from hexadecimal more easily. This is because the [[radix]] of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 24, so it takes four digits of binary to represent one digit of hexadecimal, as shown in the adjacent table.

To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:

:3A16 = 0011 10102
:E716 = 1110 01112

To convert a binary number into its hexadecimal equivalent, divide it into groups of four bits. If the number of bits isn't a multiple of four, simply insert extra '''0''' bits at the left (called [[Padding (cryptography)#Bit padding|padding]]). For example:

:10100102 = 0101 0010 grouped with padding = 5216
:110111012 = 1101 1101 grouped = DD16

To convert a hexadecimal number into its decimal equivalent, multiply the decimal equivalent of each hexadecimal digit by the corresponding power of 16 and add the resulting values:

:C0E716 = (12 × 163) + (0 × 162) + (14 × 161) + (7 × 160) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,38310

===Octal===
{{Main|Octal}}
Binary is also easily converted to the [[octal]] numeral system, since octal uses a radix of 8, which is a [[power of two]] (namely, 23, so it takes exactly three binary digits to represent an octal digit). The correspondence between octal and binary numerals is the same as for the first eight digits of [[hexadecimal]] in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth.

{| class="wikitable" style="text-align:center"
!Octal!!Binary
|-
| 0 || 000
|-
| 1 || 001
|-
| 2 || 010
|-
| 3 || 011
|-
| 4 || 100
|-
| 5 || 101
|-
| 6 || 110
|-
| 7 || 111
|}

Converting from octal to binary proceeds in the same fashion as it does for [[hexadecimal]]:

:658 = 110 1012
:178 = 001 1112

And from binary to octal:

:1011002 = 101 1002 grouped = 548
:100112 = 010 0112 grouped with padding = 238

And from octal to decimal:

:658 = (6 × 81) + (5 × 80) = (6 × 8) + (5 × 1) = 5310
:1278 = (1 × 82) + (2 × 81) + (7 × 80) = (1 × 64) + (2 × 8) + (7 × 1) = 8710

==Representing real numbers==

Non-integers can be represented by using negative powers, which are set off from the other digits by means of a radix point (called a decimal point in the decimal system). For example, the binary number 11.012 means:

{|
|'''1''' × 21 || (1 × 2 = '''2''') || plus
|-
|'''1''' × 20 || (1 × 1 = '''1''') || plus
|-
|'''0''' × 2−1 || (0 × <math> \tfrac{1}{2} </math> = '''0''') || plus
|-
|'''1''' × 2−2 || (1 × <math> \tfrac{1}{4} </math> = '''0.25''')
|}

For a total of 3.25 decimal.

All dyadic rational numbers <math>\frac{p}{2^a}</math> have a ''terminating'' binary numeral—the binary representation has a finite number of terms after the radix point. Other rational numbers have binary representation, but instead of terminating, they ''recur'', with a finite sequence of digits repeating indefinitely. For instance

:<math>\frac{1_{10}}{3_{10}} = \frac{1_2}{11_2} = 0.01010101\overline{01}\ldots\,_2 </math>

:<math>\frac{12_{10}}{17_{10}} = \frac{1100_2}{10001_2} = 0.10110100 10110100\overline{10110100}\ldots\,_2 </math>

The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in decimal. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the geometric series 2−1 + 2−2 + 2−3 + ... which is 1.

Binary numerals which neither terminate nor recur represent irrational numbers. For instance,
* 0.10100100010000100000100... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
* 1.0110101000001001111001100110011111110... is the binary representation of <math>\sqrt{2}</math>, the square root of 2, another irrational. It has no discernible pattern.

= Base 5 =
'''Quinary''' /ˈkwaɪnəri/ ('''base-5''' or '''pental''') is a numeral system with five as the base. A possible origination of a quinary system is that there are five digits on either hand.

In the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100 and sixty is written as 220.

As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two highly composite numbers (4 and 6) guarantees that many recurring fractions have relatively short periods.

Today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a sub-base system, is sexagesimal, base 60, which used 10 as a sub-base.

Each quinary digit can hold log25 (approx. 2.32) bits of information.

==Comparison to other radices==
{| class="wikitable" style="float:right; text-align:center"
|+ A quinary multiplication table
|-
| × || '''1''' || '''2''' || '''3''' || '''4''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''20'''
|-
| '''1''' || 1 || 2 || 3 || 4 || 10 || 11 || 12 || 13 || 14 || 20
|-
| '''2''' || 2 || 4 || 11 || 13 || 20 || 22 || 24 || 31 || 33 || 40
|-
| '''3''' || 3 || 11 || 14 || 22 || 30 || 33 || 41 || 44 || 102 || 110
|-
| '''4''' || 4 || 13 || 22 || 31 || 40 || 44 || 103 || 112 || 121 || 130
|-
| '''10''' || 10 || 20 || 30 || 40 || 100 || 110 || 120 || 130 || 140 || 200
|-
| '''11''' || 11 || 22 || 33 || 44 || 110 || 121 || 132 || 143 || 204 || 220
|-
| '''12''' || 12 || 24 || 41 || 103 || 120 || 132 || 144 || 211 || 223 || 240
|-
| '''13''' || 13 || 31 || 44 || 112 || 130 || 143 || 211 || 224 || 242 || 310
|-
| '''14''' || 14 || 33 || 102 || 121 || 140 || 204 || 223 || 242 || 311 || 330
|-
| '''20''' || 20 || 40 || 110 || 130 || 200 || 220 || 240 || 310 || 330 || 400
|}

{| class="wikitable"
|+ '''Numbers zero to twenty-five in standard quinary'''
|- align="center"
! Quinary
| 0 || 1 || 2 || 3 || 4 || 10 || 11 || 12 || 13 || 14 || 20 || 21 || 22
|- align="center"
! Binary
| 0 || 1 || 10 || 11 || 100 || 101 || 110 || 111 || 1000 || 1001 || 1010 || 1011 || 1100
|- align="center"
! Decimal
! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12
|- align="center"
!
|- align="center"
! Quinary
| 23 || 24 || 30 || 31 || 32 || 33 || 34 || 40 || 41 || 42 || 43 || 44 || 100
|- align="center"
! Binary
| 1101 || 1110 || 1111 || 10000 || 10001 || 10010 || 10011 || 10100 || 10101 || 10110 || 10111 || 11000 || 11001
|- align="center"
! Decimal
! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20 !! 21 !! 22 !! 23 !! 24 !! 25
|}

{| class="wikitable"
|+ '''Fractions in quinary'''
|'''Decimal''' (periodic part) ||'''Quinary''' (periodic part)
|'''Binary''' (periodic part)
|-
|1/2 = 0.5
|'''1/2''' = 0.2
|1/10 = 0.1
|-
|1/3 = 0.3
|'''1/3''' = 0.13
|1/11 = 0.01
|-
|1/4 = 0.25
|'''1/4''' = 0.1
|1/100 = 0.01
|-
|1/5 = 0.2
|'''1/10''' = 0.1
|1/101 = 0.0011
|-
|1/6 = 0.16
|'''1/11''' = 0.04
|1/110 = 0.010
|-
|1/7 = 0.142857
|'''1/12''' = 0.032412
|1/111 = 0.001
|-
|1/8 = 0.125
|'''1/13''' = 0.03
|1/1000 = 0.001
|-
|1/9 = 0.1
|'''1/14''' = 0.023421
|1/1001 = 0.000111
|-
|1/10 = 0.1
|'''1/20''' = 0.02
|1/1010 = 0.00011
|-
|1/11 = 0.09
|'''1/21''' = 0.02114
|1/1011 = 0.0001011101
|-
|1/12 = 0.083
|'''1/22''' = 0.02
|1/1100 = 0.0001
|-
|1/13 = 0.076923
|'''1/23''' = 0.0143
|1/1101 = 0.000100111011
|-
|1/14 = 0.0714285
|'''1/24''' = 0.013431
|1/1110 = 0.0001
|-
|1/15 = 0.06
|'''1/30''' = 0.013
|1/1111 = 0.0001
|-
|1/16 = 0.0625
|'''1/31''' = 0.0124
|1/10000 = 0.0001
|-
|1/17 = 0.0588235294117647
|'''1/32''' = 0.0121340243231042
|1/10001 = 0.00001111
|-
|1/18 = 0.05
|'''1/33''' = 0.011433
|1/10010 = 0.0000111
|-
|1/19 = 0.052631578947368421
|'''1/34''' = 0.011242141
|1/10011 = 0.000011010111100101
|-
|1/20 = 0.05
|'''1/40''' = 0.01
|1/10100 = 0.000011
|-
|1/21 = 0.047619
|'''1/41''' = 0.010434
|1/10101 = 0.000011
|-
|1/22 = 0.045
|'''1/42''' = 0.01032
|1/10110 = 0.00001011101
|-
|1/23 = 0.0434782608695652173913
|'''1/43''' = 0.0102041332143424031123
|1/10111 = 0.00001011001
|-
|1/24 = 0.0416
|'''1/44''' = 0.01
|1/11000 = 0.00001
|-
|1/25 = 0.04
|'''1/100''' = 0.01
|1/11001 = 0.00001010001111010111
|}

==Usage==
Many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below:

{| class="wikitable" border="1" style="text-align:center"
! Number !! Base 5 !! Numeral
|-
! style="text-align:right" | 1
| 1
| wanggany
|-
! style="text-align:right" | 2
| 2
| marrma
|-
! style="text-align:right" | 3
| 3
| lurrkun
|-
! style="text-align:right" | 4
| 4
| dambumiriw
|-
! style="text-align:right" | 5
| 10
| wanggany rulu
|-
! style="text-align:right" | 10
| 20
| marrma rulu
|-
! style="text-align:right" | 15
| 30
| lurrkun rulu
|-
! style="text-align:right" | 20
| 40
| dambumiriw rulu
|-
! style="text-align:right" | 25
| 100
| dambumirri rulu
|-
! style="text-align:right" | 50
| 200
| marrma dambumirri rulu
|-
! style="text-align:right" | 75
| 300
| lurrkun dambumirri rulu
|-
! style="text-align:right" | 100
| 400
| dambumiriw dambumirri rulu
|-
! style="text-align:right" | 125
| 1000
| dambumirri dambumirri rulu
|-
! style="text-align:right" | 625
| 10000

| dambumirri dambumirri dambumirri rulu
|}

==Biquinary==
[[File:Chinese-abacus.jpg|thumb|right|Chinese Abacus or suanpan]]
A decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are an early biquinary system. The numbers 1, 5, 10, and 50 are written as '''I''', '''V''', '''X''', and '''L''' respectively. Seven is '''VII''' and seventy is '''LXX'''. The full list is:
{| class="wikitable" style="text-align:center"
| '''I''' || '''V''' || '''X''' || '''L''' || '''C''' || '''D''' || '''M'''
|-
| 1 || 5 || 10 || 50 || 100 || 500 || 1000
|}
Note that these are not positional number systems. In theory a number such as 73 could be written as IIIXXL without ambiguity as well as LXXIII and it is still not possible to extend it beyond thousands. There is also no sign for zero. But with the introduction of inversions such as IV and IX, it was necessary to keep the order from most to least significant.

Many versions of the abacus, such as the suanpan and soroban, use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary. Units of currencies are commonly partially or wholly biquinary.

Bi-quinary coded decimal is a variant of biquinary that was used on a number of early computers including Colossus and the IBM 650 to represent decimal numbers.

= Base 10 =

The '''decimal''' numeral system (also called the '''base-ten''' positional numeral system, and occasionally called '''denary''' /ˈdiːnəri/ or '''decanary''') is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''.

A ''decimal numeral'' (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in 25.9703 or 3,1415). ''Decimal'' may also refer specifically to the digits after the decimal separator, such as in "3.14 is the approximation of <math> \pi </math> to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.

The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form {{math|''a''/10''n''}}, where {{math|''a''}} is an integer, and {{math|''n''}} is a non-negative integer.

The decimal system has been extended to ''infinite decimals'' for representing any real number, by using an infinite sequence of digits after the decimal separator (see decimal representation). In this context, the decimal numerals with a finite number of non-zero digits after the decimal separator are sometimes called ''terminating decimals''. A ''repeating decimal'' is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., 5.123144144144144... = <math> 5.123 \overline{144}</math>). An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

==Origin==
[[File:Two hand, ten fingers.jpg|thumb|right|Ten fingers on two hands, the possible origin of decimal counting]]
Many numeral systems of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the Egyptian numerals, then the Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called ''decimal fractions'' or ''decimal numbers'', for forming the ''decimal numeral system''.

== Decimal notation ==
For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "−". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; and a comma "{{math|,}}" in other countries.

For representing a non-negative number, a decimal numeral consists of
* either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer,
*:<math>a_ma_{m-1}\ldots a_0</math>
*or a decimal mark separating two sequences of digits (such as "20.70828")
::<math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math>.
If {{math|''m'' > 0}}, that is, if the first sequence contains at least two digits, it is generally assumed that the first digit {{math|''a''''m''}} is not zero. In some circumstances it may be useful to have one or more 0's on the left; this does not change the value represented by the decimal: for example, {{math|1=3.14 = 03.14 = 003.14}}. Similarly, if the final digit on the right of the decimal mark is zero—that is, if {{math|1=''b''''n'' = 0}}—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; for example, {{math|1=15 = 15.0 = 15.00}} and {{math|1=5.2 = 5.20 = 5.200}}.

For representing a negative number, a minus sign is placed before {{math|''a''''m''}}.

The numeral <math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math> represents the number
:<math>a_m10^m+a_{m-1}10^{m-1}+\cdots+a_{0}10^0+\frac{b_1}{10^1}+\frac{b_2}{10^2}+\cdots+\frac{b_n}{10^n}</math>.
The ''integer part'' or ''integral part'' of a decimal numeral is the integer written to the left of the decimal separator (see also truncation). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the ''fractional part'', which equals the difference between the numeral and its integer part.

When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example {{math|.1234}}, instead of {{math|0.1234}}). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation.

In brief, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is a positional numeral system.

== Decimal fractions ==
'''Decimal fractions''' (sometimes called '''decimal numbers''', especially in contexts involving explicit fractions) are the rational numbers that may be expressed as a fraction whose denominator is a power of ten. For example, the decimals <math>0.8, 14.89, 0.00024, 1.618, 3.14159</math> represent the fractions <math> \tfrac{8}{10}</math>, <math> \tfrac{1489}{100}</math>, <math> \tfrac{24}{100000}</math>, <math>\tfrac {1618}{1000}</math> and <math>\tfrac{314159}{100000}</math>, and are therefore decimal numbers.

More generally, a decimal with {{math|''n''}} digits after the separator represents the fraction with denominator {{math|10''n''}}, whose numerator is the integer obtained by removing the separator.

It follows that a number is a decimal fraction if and only if it has a finite decimal representation.

Expressed as a fully reduced fraction, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are
:<math>1=2^0\cdot 5^0, 2=2^1\cdot 5^0, 4=2^2\cdot 5^0, 5=2^0\cdot 5^1, 8=2^3\cdot 5^0, 10=2^1\cdot 5^1, 16=2^4\cdot 5^0, 25=2^0\cdot 5^2, \ldots</math>

==Real number approximation==

Decimal numerals do not allow an exact representation for all real numbers, e.g. for the real number <math>\pi</math>. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates the real <math>\pi</math>, being less than 10−5 off; so decimals are widely used in science, engineering and everyday life.

More precisely, for every real number {{Mvar|x}} and every positive integer {{Mvar|n}}, there are two decimals {{Mvar|''L''}} and {{Mvar|''u''}} with at most ''{{Mvar|n}}'' digits after the decimal mark such that {{Math|''L'' ≤ ''x'' ≤ ''u''}} and {{Math|1=(''u'' − ''L'') = 10−''n''}}.

Numbers are very often obtained as the result of measurement. As measurements are subject to measurement uncertainty with a known upper bound, the result of a measurement is well-represented by a decimal with {{math|''n''}} digits after the decimal mark, as soon as the absolute measurement error is bounded from above by {{Math|10−''n''}}. In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also significant figures).

==Infinite decimal expansion==

For a real number {{Mvar|x}} and an integer {{Math|''n'' ≥ 0}}, let {{Math|[''x'']''n''}} denote the (finite) decimal expansion of the greatest number that is not greater than ''{{Mvar|x}}'' that has exactly {{Mvar|n}} digits after the decimal mark. Let {{Math|''d''''i''}} denote the last digit of {{Math|[''x'']''i''}}. It is straightforward to see that {{Math|[''x'']''n''}} may be obtained by appending {{Math|''d''''n''}} to the right of {{Math|[''x'']''n''−1}}. This way one has
:{{Math|1=[''x'']''n'' = [''x'']0.''d''1''d''2...''d''''n''−1''d''''n''}},

and the difference of {{Math|[''x'']''n''−1}} and {{Math|[''x'']''n''}} amounts to
:<math>\left\vert \left [ x \right ]_n-\left [ x \right ]_{n-1} \right\vert=d_n\cdot10^{-n}<10^{-n+1}</math>,

which is either 0, if {{Math|1=''d''''n'' = 0}}, or gets arbitrarily small as ''{{Mvar|n}}'' tends to infinity. According to the definition of a limit, ''{{Mvar|x}}'' is the limit of {{Math|[''x'']''n''}} when ''{{Mvar|n}}'' tends to infinity. This is written as<math display="inline">\; x = \lim_{n\rightarrow\infty} [x]_n \;</math>or
: {{Math|1=''x'' = [''x'']0.''d''1''d''2...''d''''n''...}},
which is called an '''infinite decimal expansion''' of ''{{Mvar|x}}''.

Conversely, for any integer {{Math|[''x'']0}} and any sequence of digits<math display="inline">\;(d_n)_{n=1}^{\infty}</math> the (infinite) expression {{Math|[''x'']0.''d''1''d''2...''d''''n''...}} is an ''infinite decimal expansion'' of a real number ''{{Mvar|x}}''. This expansion is unique if neither all {{Math|''d''''n''}} are equal to 9 nor all {{Math|''d''''n''}} are equal to 0 for ''{{Mvar|n}}'' large enough (for all ''{{Mvar|n}}'' greater than some natural number {{Mvar|N}}).

If all {{Math|''d''''n''}} for {{Math|''n'' > ''N''}} equal to 9 and {{Math|1=[''x'']''n'' = [''x'']0.''d''1''d''2...''d''''n''}}, the limit of the sequence<math display="inline">\;([x]_n)_{n=1}^{\infty}</math> is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: {{Math|''d''''N''}}, by {{Math|''d''''N'' + 1}}, and replacing all subsequent 9s by 0s (see 0.999...).

Any such decimal fraction, i.e.: {{Math|1=''d''''n'' = 0}} for {{Math|''n'' > ''N''}}, may be converted to its equivalent infinite decimal expansion by replacing {{Math|''d''''N''}} by {{Math|''d''''N'' − 1}} and replacing all subsequent 0s by 9s (see 0.999...).

In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of {{Math|[''x'']''n''}}, and the other containing only 9s after some place, which is obtained by defining {{Math|[''x'']''n''}} as the greatest number that is ''less'' than {{Mvar|x}}, having exactly ''{{Mvar|n}}'' digits after the decimal mark.

=== Rational numbers ===

Long division allows computing the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a ''repeating decimal''. For example,
:<math> \tfrac{1}{81}</math> = 0. 012345679 012... (with the group 012345679 indefinitely repeating).

The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational.
{|
|-
|For example, if ''x'' is || 0.4156156156...
|-
|then 10,000''x'' is || 4156.156156156...
|-
|and 10''x'' is|| 4.156156156...
|-
|so 10,000''x'' − 10''x'', i.e. 9,990''x'', is|| 4152.000000000...
|-
|and ''x'' is|| <math> \tfrac{4152}{9990}</math>
|}
or, dividing both numerator and denominator by 6, <math> \tfrac{692}{1665}</math>.

== Decimal computation ==

[[File:Decimal multiplication table.JPG|thumb|right|300px|Diagram of the world's earliest known multiplication table (c. 305 BCE) from the Warring States period]]

Most modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as the ENIAC or the IBM 650, used decimal representation internally).
For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems.

For most purposes, however, binary values are converted to or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)

Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of binary-coded decimal, especially in database implementations, but there are other decimal representations in use (including decimal floating point such as in newer revisions of the IEEE 754|IEEE 754 Standard for Floating-Point Arithmetic).

Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of <math>10</math> have no finite binary fractional representation; and is generally impossible for multiplication (or division). See Arbitrary-precision arithmetic for exact calculations.

= Licensing =
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Radix Radix, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Binary_number Binary number, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Quinary Quinary, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Decimal Decimal, Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T18:29:03Z

Khanh:

== Formal logical systems ==
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.

=== First-order logic ===
'''First-order logic''' is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.

Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.

Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The '''first incompleteness theorem''' states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The '''second incompleteness theorem''' states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

=== Other classical logics ===
Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''fixed-point logics''' that allow inductive definitions, like one writes for primitive recursive functions.

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

=== Nonclassical and modal logic ===
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

=== Algebraic logic ===
Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.

== Problem-solving strategies ==
Problem-solving strategies are the steps that one would use to find the problems that are in the way to getting to one's own goal. Some refer to this as the "problem-solving cycle".

In this cycle one will acknowledge, recognize the problem, define the problem, develop a strategy to fix the problem, organize the knowledge of the problem cycle, figure out the resources at the user's disposal, monitor one's progress, and evaluate the solution for accuracy. The reason it is called a cycle is that once one is completed with a problem another will usually pop up.

'''Insight''' is the sudden solution to a long-vexing '''problem''', a sudden recognition of a new idea, or a sudden understanding of a complex situation, an ''Aha!'' moment. Solutions found through '''insight''' are often more accurate than those found through step-by-step analysis. To solve more problems at a faster rate, insight is necessary for selecting productive moves at different stages of the problem-solving cycle. This problem-solving strategy pertains specifically to problems referred to as insight problem. Unlike Newell and Simon's formal definition of move problems, there has not been a generally agreed upon definition of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011).

Blanchard-Fields looks at problem solving from one of two facets. The first looking at those problems that only have one solution (like mathematical problems, or fact-based questions) which are grounded in psychometric intelligence. The other is socioemotional in nature and have answers that change constantly (like what's your favorite color or what you should get someone for Christmas).

The following techniques are usually called ''problem-solving strategies''
* Abstraction: solving the problem in a model of the system before applying it to the real system
* Analogy: using a solution that solves an analogous problem
* Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
* Critical thinking
* Divide and conquer: breaking down a large, complex problem into smaller, solvable problems
* Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
* Lateral thinking: approaching solutions indirectly and creatively
* Means-ends analysis: choosing an action at each step to move closer to the goal
* Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new
* Morphological analysis: assessing the output and interactions of an entire system
* Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
* Reduction: transforming the problem into another problem for which solutions exist
* Research: employing existing ideas or adapting existing solutions to similar problems
* Root cause analysis: identifying the cause of a problem
* Trial-and-error: testing possible solutions until the right one is found

== Multiple forms of representation ==
=== Concrete models===
====Base ten blocks====
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

====Pattern blocks====
[[File:Wooden pattern blocks dodecagon.JPG|thumb|One of the ways of making a dodecagon with pattern blocks]]
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.

Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.

Pattern blocks can also serve to provide students with an understanding of fractions. Because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.

Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

====Unifix® Cubes====
[[File:Linking cm cubes 2.JPG|thumb|Interlocking centimeter cubes]]
Unifix® Cubes are interlocking cubes that are just under 2 centimeters on each side. The cubes connect with each other from one side. Once connected, Unifix® Cubes can be turned to form a vertical Unifix® "tower," or horizontally to form a Unifix® "train".

Other interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities.

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence,
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue..
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.

Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume.

====Tiles====
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).

Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.

These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

====Number lines====
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

====Cuisenaire rods====
[[File:Cuisenaire ten.JPG|thumb|Cuisenaire rods used to illustrate the factors of ten]]
'''Cuisenaire rods''' are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods ''réglettes''.

According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."

;Use in mathematics teaching
The rods are used in teaching a variety of mathematical ideas, and with a wide age range of learners. Topics they are used for include:
* Counting, sequences, patterns and algebraic reasoning
* Addition and subtraction (additive reasoning)
* Multiplication and division (multiplicative reasoning)
* Fractions, ratio and proportion
* Modular arithmetic leading to group theory

=== Pictures ===
[[File:Titanic casualties.svg|thumb|300px|alt=table with boxes instead of numbers, the amounts and sizes of boxes represent amounts of people|A compound pictogram showing the breakdown of the survivors and deaths of the maiden voyage of the RMS Titanic by class and age/gender.]]

Pictograms are charts in which icons represent numbers to make it more interesting and easier to understand. A key is often included to indicate what each icon represents. All icons must be of the same size, but a fraction of an icon can be used to show the respective fraction of that amount.

For example, the following table:
{| class="wikitable"
! Day !! Letters sent
|-
| Monday || 10
|-
| Tuesday || 17
|-
| Wednesday || 29
|-
| Thursday || 41
|-
| Friday || 18
|}
can be graphed as follows:
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+
! Day !! Letters sent
|-
| Monday || [[File:Email Silk.svg|alt=one envelope]]
|-
| Tuesday || [[File:Email Silk.svg|alt=one envelope]] [[File:Image from the Silk icon theme by Mark James half left.svg|alt=and a half]]
|-
| Wednesday || [[File:Email Silk.svg|alt=three envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Thursday || [[File:Email Silk.svg|alt=four envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Friday || [[File:Email Silk.svg|alt=two envelopes]] [[File:Email Silk.svg|alt=]]
|}
Key: [[File:Email Silk.svg|alt=one envelope]] = 10 letters

As the values are rounded to the nearest 5 letters, the second icon on Tuesday is the left half of the original.

=== Diagrams ===
A '''diagram''' is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram.

====Gallery of diagram types====

There are at least the following types of diagrams:

* Logical or conceptual diagrams, which take a collection of items and relationships between them, and express them by giving each item a 2D position, while the relationships are expressed as connections between the items or overlaps between the items, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Tree Example.png|Tree diagram
File:Neural network.svg|Network diagram
File:LampFlowchart.svg|Flowchart
File:Set intersection.svg|Venn diagram
File:Alphagraphen.png|Existential graph
</gallery>
|}

* Quantitative diagrams, which display a relationship between two variables that take either discrete or a continuous range of values; for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Histogram example.svg|Histogram
File:Graphtestone.svg|Bar graph
File:Zusammensetzung Shampoo.svg|Pie chart
File:Hyperbolic Cosine.svg|Function graph
File:R-car stopping distances 1920.svg|Scatter plot
File:Hanger Diagram.png|Hanger diagram.
</gallery>
|}

* Schematics and other types of diagrams, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Train schedule of Sanin Line, Japan, 1949-09-15, part.png|Time–distance diagram
File:Gear pump exploded.png|Exploded view
File:US 2000 census population density map by state.svg|Population density map
File:Pioneer plaque.svg|Pioneer plaque
File:Automotive diagrams 01 En.png|Three-dimensional diagram
</gallery>
|}

=== Table ===
[[File:Table-sample-appearance-default-params-values-01.gif|thumb|300px|An example table rendered in a web browser using HTML.]]
A '''table''' is an arrangement of information or data, typically in rows and columns, or possibly in a more complex structure. Tables are widely used in communication, research, and data analysis. Tables appear in print media, handwritten notes, computer software, architectural ornamentation, traffic signs, and many other places. The precise conventions and terminology for describing tables vary depending on the context. Further, tables differ significantly in variety, structure, flexibility, notation, representation and use. Information or data conveyed in table form is said to be in '''tabular''' format (adjective). In books and technical articles, tables are typically presented apart from the main text in numbered and captioned floating blocks.

==== Basic description ====
A table consists of an ordered arrangement of '''rows''' and '''columns'''. This is a simplified description of the most basic kind of table. Certain considerations follow from this simplified description:

* the term '''row''' has several common synonyms (e.g., record, k-tuple, n-tuple, vector);
* the term '''column''' has several common synonyms (e.g., field, parameter, property, attribute, stanchion);
* a column is usually identified by a name;
* a column name can consist of a word, phrase or a numerical index;
* the intersection of a row and a column is called a cell.

The elements of a table may be grouped, segmented, or arranged in many different ways, and even nested recursively. Additionally, a table may include metadata, annotations, a header, a footer or other ancillary features.

; Simple table
The following illustrates a simple table with three columns and nine rows. The first row is not counted, because it is only used to display the column names. This is called a "header row".

{| class="wikitable" style="text-align:center"
|+Age table
|-
! First name !! Last name !! Age
|-
| Tinu || Elejogun|| 14
|-
| Javier || Zapata || 28
|-
| Lily || McGarrett || 18
|-
| Olatunkbo || Chijiaku || 22
|-
| Adrienne || Anthoula || 22
|-
| Axelia|| Athanasios || 22
|-
| Jon-Kabat || Zinn || 22
|-
| Thabang|| Mosoa||15
|-
| Kgaogelo|| Mosoa||11
|-
|}

; Multi-dimensional table
[[File:Rollup table.png|thumb|An example of a table containing rows with summary information. The summary information consists of subtotals that are combined from previous rows within the same column.]]

The concept of '''dimension''' is also a part of basic terminology. Any "simple" table can be represented as a "multi-dimensional"
table by normalizing the data values into ordered hierarchies. A common example of such a table is a multiplication table.

{| class="wikitable" style="text-align:center"
|+ style="white-space:nowrap" |Multiplication table
|-
!×!! 1 !! 2 !! 3
|-
! 1
| 1 || 2 || 3
|-
! 2
| 2 || 4 || 6
|-
! 3
| 3 || 6 || 9
|}

In multi-dimensional tables, each cell in the body of the table (and the value of that cell) relates to the values at the beginnings of the column (i.e. the header), the row, and other structures in more complex tables. This is an injective relation: each combination of the values of the headers row (row 0, for lack of a better term) and the headers column (column 0 for lack of a better term) is related to a unique cell in
the table:

* Column 1 and row 1 will only correspond to cell (1,1);
* Column 1 and row 2 will only correspond to cell (2,1) etc.

The first column often presents information dimension description by which the rest of the table is navigated. This column is called "stub column". Tables may contain three or multiple dimensions and can be classified by the number of dimensions. Multi-dimensional tables may have super-rows - rows that describe additional dimensions for the rows that are presented below that row and are usually grouped in a tree-like structure. This structure is typically visually presented with an appropriate number of white spaces in front of each stub's label.

In literature tables often present numerical values, cumulative statistics, categorical values, and at times parallel descriptions in form of text. They can condense large amount of information to a limited space and therefore they are popular in scientific literature in many fields of study.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Mathematical_logic Mathematical logic, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Problem_solving Problem solving, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Manipulative_(mathematics_education) Manipulative (mathematics education), Wikipedia] under a CC BY-Sa license
* [https://en.wikipedia.org/wiki/Cuisenaire_rods Cuisenaire rods, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Pictogram Pictogram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Diagram Diagram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Table_(information) Table (information), Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T18:19:42Z

Khanh:

== Formal logical systems ==
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.

=== First-order logic ===
'''First-order logic''' is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.

Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.

Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The '''first incompleteness theorem''' states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The '''second incompleteness theorem''' states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

=== Other classical logics ===
Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''fixed-point logics''' that allow inductive definitions, like one writes for primitive recursive functions.

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

=== Nonclassical and modal logic ===
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

=== Algebraic logic ===
Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.

== Problem-solving strategies ==
Problem-solving strategies are the steps that one would use to find the problems that are in the way to getting to one's own goal. Some refer to this as the "problem-solving cycle".

In this cycle one will acknowledge, recognize the problem, define the problem, develop a strategy to fix the problem, organize the knowledge of the problem cycle, figure out the resources at the user's disposal, monitor one's progress, and evaluate the solution for accuracy. The reason it is called a cycle is that once one is completed with a problem another will usually pop up.

'''Insight''' is the sudden solution to a long-vexing '''problem''', a sudden recognition of a new idea, or a sudden understanding of a complex situation, an ''Aha!'' moment. Solutions found through '''insight''' are often more accurate than those found through step-by-step analysis. To solve more problems at a faster rate, insight is necessary for selecting productive moves at different stages of the problem-solving cycle. This problem-solving strategy pertains specifically to problems referred to as insight problem. Unlike Newell and Simon's formal definition of move problems, there has not been a generally agreed upon definition of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011).

Blanchard-Fields looks at problem solving from one of two facets. The first looking at those problems that only have one solution (like mathematical problems, or fact-based questions) which are grounded in psychometric intelligence. The other is socioemotional in nature and have answers that change constantly (like what's your favorite color or what you should get someone for Christmas).

The following techniques are usually called ''problem-solving strategies''
* Abstraction: solving the problem in a model of the system before applying it to the real system
* Analogy: using a solution that solves an analogous problem
* Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
* Critical thinking
* Divide and conquer: breaking down a large, complex problem into smaller, solvable problems
* Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
* Lateral thinking: approaching solutions indirectly and creatively
* Means-ends analysis: choosing an action at each step to move closer to the goal
* Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new
* Morphological analysis: assessing the output and interactions of an entire system
* Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
* Reduction: transforming the problem into another problem for which solutions exist
* Research: employing existing ideas or adapting existing solutions to similar problems
* Root cause analysis: identifying the cause of a problem
* Trial-and-error: testing possible solutions until the right one is found

== Multiple forms of representation ==
=== Concrete models===
====Base ten blocks====
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

====Pattern blocks====
[[File:Wooden pattern blocks dodecagon.JPG|thumb|One of the ways of making a dodecagon with pattern blocks]]
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.

Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.

Pattern blocks can also serve to provide students with an understanding of fractions. Because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.

Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

====Unifix® Cubes====
[[File:Linking cm cubes 2.JPG|thumb|Interlocking centimeter cubes]]
Unifix® Cubes are interlocking cubes that are just under 2 centimeters on each side. The cubes connect with each other from one side. Once connected, Unifix® Cubes can be turned to form a vertical Unifix® "tower," or horizontally to form a Unifix® "train".

Other interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities.

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence,
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue..
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.

Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume.

====Tiles====
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).

Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.

These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

====Number lines====
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

====Cuisenaire rods====
[[File:Cuisenaire ten.JPG|thumb|Cuisenaire rods used to illustrate the factors of ten]]
'''Cuisenaire rods''' are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods ''réglettes''.

According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."

;Use in mathematics teaching
The rods are used in teaching a variety of mathematical ideas, and with a wide age range of learners. Topics they are used for include:
* Counting, sequences, patterns and algebraic reasoning
* Addition and subtraction (additive reasoning)
* Multiplication and division (multiplicative reasoning)
* Fractions, ratio and proportion
* Modular arithmetic leading to group theory

=== Pictures ===
[[File:Titanic casualties.svg|thumb|300px|alt=table with boxes instead of numbers, the amounts and sizes of boxes represent amounts of people|A compound pictogram showing the breakdown of the survivors and deaths of the maiden voyage of the RMS Titanic by class and age/gender.]]

Pictograms are charts in which icons represent numbers to make it more interesting and easier to understand. A key is often included to indicate what each icon represents. All icons must be of the same size, but a fraction of an icon can be used to show the respective fraction of that amount.

For example, the following table:
{| class="wikitable"
! Day !! Letters sent
|-
| Monday || 10
|-
| Tuesday || 17
|-
| Wednesday || 29
|-
| Thursday || 41
|-
| Friday || 18
|}
can be graphed as follows:
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+
! Day !! Letters sent
|-
| Monday || [[File:Email Silk.svg|alt=one envelope]]
|-
| Tuesday || [[File:Email Silk.svg|alt=one envelope]] [[File:Image from the Silk icon theme by Mark James half left.svg|alt=and a half]]
|-
| Wednesday || [[File:Email Silk.svg|alt=three envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Thursday || [[File:Email Silk.svg|alt=four envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Friday || [[File:Email Silk.svg|alt=two envelopes]] [[File:Email Silk.svg|alt=]]
|}
Key: [[File:Email Silk.svg|alt=one envelope]] = 10 letters

As the values are rounded to the nearest 5 letters, the second icon on Tuesday is the left half of the original.

=== Diagrams ===
A '''diagram''' is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram.

====Gallery of diagram types====

There are at least the following types of diagrams:

* Logical or conceptual diagrams, which take a collection of items and relationships between them, and express them by giving each item a 2D position, while the relationships are expressed as connections between the items or overlaps between the items, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Tree Example.png|Tree diagram
File:Neural network.svg|Network diagram
File:LampFlowchart.svg|Flowchart
File:Set intersection.svg|Venn diagram
File:Alphagraphen.png|Existential graph
</gallery>
|}

* Quantitative diagrams, which display a relationship between two variables that take either discrete or a continuous range of values; for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Histogram example.svg|Histogram
File:Graphtestone.svg|Bar graph
File:Zusammensetzung Shampoo.svg|Pie chart
File:Hyperbolic Cosine.svg|Function graph
File:R-car stopping distances 1920.svg|Scatter plot
File:Hanger Diagram.png|Hanger diagram.
</gallery>
|}

* Schematics and other types of diagrams, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Train schedule of Sanin Line, Japan, 1949-09-15, part.png|Time–distance diagram
File:Gear pump exploded.png|Exploded view
File:US 2000 census population density map by state.svg|Population density map
File:Pioneer plaque.svg|Pioneer plaque
File:Automotive diagrams 01 En.png|Three-dimensional diagram
</gallery>
|}

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Mathematical_logic Mathematical logic, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Problem_solving Problem solving, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Manipulative_(mathematics_education) Manipulative (mathematics education), Wikipedia] under a CC BY-Sa license
* [https://en.wikipedia.org/wiki/Cuisenaire_rods Cuisenaire rods, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Pictogram Pictogram, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Diagram Diagram, Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T18:15:52Z

Khanh: /* Pictures */

== Formal logical systems ==
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.

=== First-order logic ===
'''First-order logic''' is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.

Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.

Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The '''first incompleteness theorem''' states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The '''second incompleteness theorem''' states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

=== Other classical logics ===
Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''fixed-point logics''' that allow inductive definitions, like one writes for primitive recursive functions.

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

=== Nonclassical and modal logic ===
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

=== Algebraic logic ===
Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.

== Problem-solving strategies ==
Problem-solving strategies are the steps that one would use to find the problems that are in the way to getting to one's own goal. Some refer to this as the "problem-solving cycle".

In this cycle one will acknowledge, recognize the problem, define the problem, develop a strategy to fix the problem, organize the knowledge of the problem cycle, figure out the resources at the user's disposal, monitor one's progress, and evaluate the solution for accuracy. The reason it is called a cycle is that once one is completed with a problem another will usually pop up.

'''Insight''' is the sudden solution to a long-vexing '''problem''', a sudden recognition of a new idea, or a sudden understanding of a complex situation, an ''Aha!'' moment. Solutions found through '''insight''' are often more accurate than those found through step-by-step analysis. To solve more problems at a faster rate, insight is necessary for selecting productive moves at different stages of the problem-solving cycle. This problem-solving strategy pertains specifically to problems referred to as insight problem. Unlike Newell and Simon's formal definition of move problems, there has not been a generally agreed upon definition of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011).

Blanchard-Fields looks at problem solving from one of two facets. The first looking at those problems that only have one solution (like mathematical problems, or fact-based questions) which are grounded in psychometric intelligence. The other is socioemotional in nature and have answers that change constantly (like what's your favorite color or what you should get someone for Christmas).

The following techniques are usually called ''problem-solving strategies''
* Abstraction: solving the problem in a model of the system before applying it to the real system
* Analogy: using a solution that solves an analogous problem
* Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
* Critical thinking
* Divide and conquer: breaking down a large, complex problem into smaller, solvable problems
* Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
* Lateral thinking: approaching solutions indirectly and creatively
* Means-ends analysis: choosing an action at each step to move closer to the goal
* Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new
* Morphological analysis: assessing the output and interactions of an entire system
* Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
* Reduction: transforming the problem into another problem for which solutions exist
* Research: employing existing ideas or adapting existing solutions to similar problems
* Root cause analysis: identifying the cause of a problem
* Trial-and-error: testing possible solutions until the right one is found

== Multiple forms of representation ==
=== Concrete models===
====Base ten blocks====
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

====Pattern blocks====
[[File:Wooden pattern blocks dodecagon.JPG|thumb|One of the ways of making a dodecagon with pattern blocks]]
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.

Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.

Pattern blocks can also serve to provide students with an understanding of fractions. Because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.

Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

====Unifix® Cubes====
[[File:Linking cm cubes 2.JPG|thumb|Interlocking centimeter cubes]]
Unifix® Cubes are interlocking cubes that are just under 2 centimeters on each side. The cubes connect with each other from one side. Once connected, Unifix® Cubes can be turned to form a vertical Unifix® "tower," or horizontally to form a Unifix® "train".

Other interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities.

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence,
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue..
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.

Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume.

====Tiles====
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).

Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.

These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

====Number lines====
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

====Cuisenaire rods====
[[File:Cuisenaire ten.JPG|thumb|Cuisenaire rods used to illustrate the factors of ten]]
'''Cuisenaire rods''' are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods ''réglettes''.

According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."

;Use in mathematics teaching
The rods are used in teaching a variety of mathematical ideas, and with a wide age range of learners. Topics they are used for include:
* Counting, sequences, patterns and algebraic reasoning
* Addition and subtraction (additive reasoning)
* Multiplication and division (multiplicative reasoning)
* Fractions, ratio and proportion
* Modular arithmetic leading to group theory

=== Pictures ===
[[File:Titanic casualties.svg|thumb|300px|alt=table with boxes instead of numbers, the amounts and sizes of boxes represent amounts of people|A compound pictogram showing the breakdown of the survivors and deaths of the maiden voyage of the RMS Titanic by class and age/gender.]]

In statistics, pictograms are charts in which icons represent numbers to make it more interesting and easier to understand. A key is often included to indicate what each icon represents. All icons must be of the same size, but a fraction of an icon can be used to show the respective fraction of that amount.

For example, the following table:
{| class="wikitable"
! Day !! Letters sent
|-
| Monday || 10
|-
| Tuesday || 17
|-
| Wednesday || 29
|-
| Thursday || 41
|-
| Friday || 18
|}
can be graphed as follows:
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+
! Day !! Letters sent
|-
| Monday || [[File:Email Silk.svg|alt=one envelope]]
|-
| Tuesday || [[File:Email Silk.svg|alt=one envelope]] [[File:Image from the Silk icon theme by Mark James half left.svg|alt=and a half]]
|-
| Wednesday || [[File:Email Silk.svg|alt=three envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Thursday || [[File:Email Silk.svg|alt=four envelopes]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]] [[File:Email Silk.svg|alt=]]
|-
| Friday || [[File:Email Silk.svg|alt=two envelopes]] [[File:Email Silk.svg|alt=]]
|}
Key: [[File:Email Silk.svg|alt=one envelope]] = 10 letters

As the values are rounded to the nearest 5 letters, the second icon on Tuesday is the left half of the original.

=== Diagrams ===
A '''diagram''' is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram.

====Gallery of diagram types====

There are at least the following types of diagrams:

* Logical or conceptual diagrams, which take a collection of items and relationships between them, and express them by giving each item a 2D position, while the relationships are expressed as connections between the items or overlaps between the items, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Tree Example.png|Tree diagram
File:Neural network.svg|Network diagram
File:LampFlowchart.svg|Flowchart
File:Set intersection.svg|Venn diagram
File:Alphagraphen.png|Existential graph
</gallery>
|}

* Quantitative diagrams, which display a relationship between two variables that take either discrete or a continuous range of values; for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Histogram example.svg|Histogram
File:Graphtestone.svg|Bar graph
File:Zusammensetzung Shampoo.svg|Pie chart
File:Hyperbolic Cosine.svg|Function graph
File:R-car stopping distances 1920.svg|Scatter plot
File:Hanger Diagram.png|Hanger diagram.
</gallery>
|}

* Schematics and other types of diagrams, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Train schedule of Sanin Line, Japan, 1949-09-15, part.png|Time–distance diagram
File:Gear pump exploded.png|Exploded view
File:US 2000 census population density map by state.svg|Population density map
File:Pioneer plaque.svg|Pioneer plaque
File:Automotive diagrams 01 En.png|Three-dimensional diagram
</gallery>
|}

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Mathematical_logic Mathematical logic, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Problem_solving Problem solving, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Manipulative_(mathematics_education) Manipulative (mathematics education), Wikipedia] under a CC BY-Sa license
* [https://en.wikipedia.org/wiki/Cuisenaire_rods Cuisenaire rods, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Diagram Diagram, Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T18:05:25Z

Khanh: /* Diagrams */

== Formal logical systems ==
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.

=== First-order logic ===
'''First-order logic''' is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.

Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.

Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The '''first incompleteness theorem''' states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The '''second incompleteness theorem''' states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

=== Other classical logics ===
Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''fixed-point logics''' that allow inductive definitions, like one writes for primitive recursive functions.

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

=== Nonclassical and modal logic ===
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

=== Algebraic logic ===
Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.

== Problem-solving strategies ==
Problem-solving strategies are the steps that one would use to find the problems that are in the way to getting to one's own goal. Some refer to this as the "problem-solving cycle".

In this cycle one will acknowledge, recognize the problem, define the problem, develop a strategy to fix the problem, organize the knowledge of the problem cycle, figure out the resources at the user's disposal, monitor one's progress, and evaluate the solution for accuracy. The reason it is called a cycle is that once one is completed with a problem another will usually pop up.

'''Insight''' is the sudden solution to a long-vexing '''problem''', a sudden recognition of a new idea, or a sudden understanding of a complex situation, an ''Aha!'' moment. Solutions found through '''insight''' are often more accurate than those found through step-by-step analysis. To solve more problems at a faster rate, insight is necessary for selecting productive moves at different stages of the problem-solving cycle. This problem-solving strategy pertains specifically to problems referred to as insight problem. Unlike Newell and Simon's formal definition of move problems, there has not been a generally agreed upon definition of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011).

Blanchard-Fields looks at problem solving from one of two facets. The first looking at those problems that only have one solution (like mathematical problems, or fact-based questions) which are grounded in psychometric intelligence. The other is socioemotional in nature and have answers that change constantly (like what's your favorite color or what you should get someone for Christmas).

The following techniques are usually called ''problem-solving strategies''
* Abstraction: solving the problem in a model of the system before applying it to the real system
* Analogy: using a solution that solves an analogous problem
* Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
* Critical thinking
* Divide and conquer: breaking down a large, complex problem into smaller, solvable problems
* Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
* Lateral thinking: approaching solutions indirectly and creatively
* Means-ends analysis: choosing an action at each step to move closer to the goal
* Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new
* Morphological analysis: assessing the output and interactions of an entire system
* Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
* Reduction: transforming the problem into another problem for which solutions exist
* Research: employing existing ideas or adapting existing solutions to similar problems
* Root cause analysis: identifying the cause of a problem
* Trial-and-error: testing possible solutions until the right one is found

== Multiple forms of representation ==
=== Concrete models===
====Base ten blocks====
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

====Pattern blocks====
[[File:Wooden pattern blocks dodecagon.JPG|thumb|One of the ways of making a dodecagon with pattern blocks]]
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.

Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.

Pattern blocks can also serve to provide students with an understanding of fractions. Because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.

Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

====Unifix® Cubes====
[[File:Linking cm cubes 2.JPG|thumb|Interlocking centimeter cubes]]
Unifix® Cubes are interlocking cubes that are just under 2 centimeters on each side. The cubes connect with each other from one side. Once connected, Unifix® Cubes can be turned to form a vertical Unifix® "tower," or horizontally to form a Unifix® "train".

Other interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities.

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence,
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue..
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.

Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume.

====Tiles====
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).

Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.

These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

====Number lines====
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

====Cuisenaire rods====
[[File:Cuisenaire ten.JPG|thumb|Cuisenaire rods used to illustrate the factors of ten]]
'''Cuisenaire rods''' are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods ''réglettes''.

According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."

;Use in mathematics teaching
The rods are used in teaching a variety of mathematical ideas, and with a wide age range of learners. Topics they are used for include:
* Counting, sequences, patterns and algebraic reasoning
* Addition and subtraction (additive reasoning)
* Multiplication and division (multiplicative reasoning)
* Fractions, ratio and proportion
* Modular arithmetic leading to group theory

=== Pictures ===

=== Diagrams ===
A '''diagram''' is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram.

====Gallery of diagram types====

There are at least the following types of diagrams:

* Logical or conceptual diagrams, which take a collection of items and relationships between them, and express them by giving each item a 2D position, while the relationships are expressed as connections between the items or overlaps between the items, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Tree Example.png|Tree diagram
File:Neural network.svg|Network diagram
File:LampFlowchart.svg|Flowchart
File:Set intersection.svg|Venn diagram
File:Alphagraphen.png|Existential graph
</gallery>
|}

* Quantitative diagrams, which display a relationship between two variables that take either discrete or a continuous range of values; for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Histogram example.svg|Histogram
File:Graphtestone.svg|Bar graph
File:Zusammensetzung Shampoo.svg|Pie chart
File:Hyperbolic Cosine.svg|Function graph
File:R-car stopping distances 1920.svg|Scatter plot
File:Hanger Diagram.png|Hanger diagram.
</gallery>
|}

* Schematics and other types of diagrams, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Train schedule of Sanin Line, Japan, 1949-09-15, part.png|Time–distance diagram
File:Gear pump exploded.png|Exploded view
File:US 2000 census population density map by state.svg|Population density map
File:Pioneer plaque.svg|Pioneer plaque
File:Automotive diagrams 01 En.png|Three-dimensional diagram
</gallery>
|}

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Mathematical_logic Mathematical logic, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Problem_solving Problem solving, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Manipulative_(mathematics_education) Manipulative (mathematics education), Wikipedia] under a CC BY-Sa license
* [https://en.wikipedia.org/wiki/Cuisenaire_rods Cuisenaire rods, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Diagram Diagram, Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T18:03:51Z

Khanh:

== Formal logical systems ==
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.

=== First-order logic ===
'''First-order logic''' is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.

Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.

Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The '''first incompleteness theorem''' states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The '''second incompleteness theorem''' states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

=== Other classical logics ===
Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''fixed-point logics''' that allow inductive definitions, like one writes for primitive recursive functions.

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

=== Nonclassical and modal logic ===
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

=== Algebraic logic ===
Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.

== Problem-solving strategies ==
Problem-solving strategies are the steps that one would use to find the problems that are in the way to getting to one's own goal. Some refer to this as the "problem-solving cycle".

In this cycle one will acknowledge, recognize the problem, define the problem, develop a strategy to fix the problem, organize the knowledge of the problem cycle, figure out the resources at the user's disposal, monitor one's progress, and evaluate the solution for accuracy. The reason it is called a cycle is that once one is completed with a problem another will usually pop up.

'''Insight''' is the sudden solution to a long-vexing '''problem''', a sudden recognition of a new idea, or a sudden understanding of a complex situation, an ''Aha!'' moment. Solutions found through '''insight''' are often more accurate than those found through step-by-step analysis. To solve more problems at a faster rate, insight is necessary for selecting productive moves at different stages of the problem-solving cycle. This problem-solving strategy pertains specifically to problems referred to as insight problem. Unlike Newell and Simon's formal definition of move problems, there has not been a generally agreed upon definition of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011).

Blanchard-Fields looks at problem solving from one of two facets. The first looking at those problems that only have one solution (like mathematical problems, or fact-based questions) which are grounded in psychometric intelligence. The other is socioemotional in nature and have answers that change constantly (like what's your favorite color or what you should get someone for Christmas).

The following techniques are usually called ''problem-solving strategies''
* Abstraction: solving the problem in a model of the system before applying it to the real system
* Analogy: using a solution that solves an analogous problem
* Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
* Critical thinking
* Divide and conquer: breaking down a large, complex problem into smaller, solvable problems
* Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
* Lateral thinking: approaching solutions indirectly and creatively
* Means-ends analysis: choosing an action at each step to move closer to the goal
* Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new
* Morphological analysis: assessing the output and interactions of an entire system
* Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
* Reduction: transforming the problem into another problem for which solutions exist
* Research: employing existing ideas or adapting existing solutions to similar problems
* Root cause analysis: identifying the cause of a problem
* Trial-and-error: testing possible solutions until the right one is found

== Multiple forms of representation ==
=== Concrete models===
====Base ten blocks====
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

====Pattern blocks====
[[File:Wooden pattern blocks dodecagon.JPG|thumb|One of the ways of making a dodecagon with pattern blocks]]
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.

Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.

Pattern blocks can also serve to provide students with an understanding of fractions. Because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.

Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

====Unifix® Cubes====
[[File:Linking cm cubes 2.JPG|thumb|Interlocking centimeter cubes]]
Unifix® Cubes are interlocking cubes that are just under 2 centimeters on each side. The cubes connect with each other from one side. Once connected, Unifix® Cubes can be turned to form a vertical Unifix® "tower," or horizontally to form a Unifix® "train".

Other interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities.

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence,
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue..
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.

Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume.

====Tiles====
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).

Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.

These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

====Number lines====
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

====Cuisenaire rods====
[[File:Cuisenaire ten.JPG|thumb|Cuisenaire rods used to illustrate the factors of ten]]
'''Cuisenaire rods''' are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods ''réglettes''.

According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."

;Use in mathematics teaching
The rods are used in teaching a variety of mathematical ideas, and with a wide age range of learners. Topics they are used for include:
* Counting, sequences, patterns and algebraic reasoning
* Addition and subtraction (additive reasoning)
* Multiplication and division (multiplicative reasoning)
* Fractions, ratio and proportion
* Modular arithmetic leading to group theory

=== Pictures ===

=== Diagrams ===
A '''diagram''' is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram.

;Gallery of diagram types

There are at least the following types of diagrams:

* Logical or conceptual diagrams, which take a collection of items and relationships between them, and express them by giving each item a 2D position, while the relationships are expressed as connections between the items or overlaps between the items, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Tree Example.png|Tree diagram
File:Neural network.svg|Network diagram
File:LampFlowchart.svg|Flowchart
File:Set intersection.svg|Venn diagram
File:Alphagraphen.png|Existential graph
</gallery>
|}

* Quantitative diagrams, which display a relationship between two variables that take either discrete or a continuous range of values; for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Histogram example.svg|Histogram
File:Graphtestone.svg|Bar graph
File:Zusammensetzung Shampoo.svg|Pie chart
File:Hyperbolic Cosine.svg|Function graph
File:R-car stopping distances 1920.svg|Scatter plot
File:Hanger Diagram.png|Hanger diagram.
</gallery>
|}

* Schematics and other types of diagrams, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Train schedule of Sanin Line, Japan, 1949-09-15, part.png|Time–distance diagram
File:Gear pump exploded.png|Exploded view
File:US 2000 census population density map by state.svg|Population density map
File:Pioneer plaque.svg|Pioneer plaque
File:Automotive diagrams 01 En.png|Three-dimensional diagram
</gallery>
|}

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Mathematical_logic Mathematical logic, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Problem_solving Problem solving, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Manipulative_(mathematics_education) Manipulative (mathematics education), Wikipedia] under a CC BY-Sa license
* [https://en.wikipedia.org/wiki/Cuisenaire_rods Cuisenaire rods, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Diagram Diagram, Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T17:51:41Z

Khanh:

== Formal logical systems ==
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.

=== First-order logic ===
'''First-order logic''' is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.

Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.

Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The '''first incompleteness theorem''' states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.

Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The '''second incompleteness theorem''' states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.

=== Other classical logics ===
Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''fixed-point logics''' that allow inductive definitions, like one writes for primitive recursive functions.

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

=== Nonclassical and modal logic ===
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

=== Algebraic logic ===
Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.

== Problem-solving strategies ==
Problem-solving strategies are the steps that one would use to find the problems that are in the way to getting to one's own goal. Some refer to this as the "problem-solving cycle".

In this cycle one will acknowledge, recognize the problem, define the problem, develop a strategy to fix the problem, organize the knowledge of the problem cycle, figure out the resources at the user's disposal, monitor one's progress, and evaluate the solution for accuracy. The reason it is called a cycle is that once one is completed with a problem another will usually pop up.

'''Insight''' is the sudden solution to a long-vexing '''problem''', a sudden recognition of a new idea, or a sudden understanding of a complex situation, an ''Aha!'' moment. Solutions found through '''insight''' are often more accurate than those found through step-by-step analysis. To solve more problems at a faster rate, insight is necessary for selecting productive moves at different stages of the problem-solving cycle. This problem-solving strategy pertains specifically to problems referred to as insight problem. Unlike Newell and Simon's formal definition of move problems, there has not been a generally agreed upon definition of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011).

Blanchard-Fields looks at problem solving from one of two facets. The first looking at those problems that only have one solution (like mathematical problems, or fact-based questions) which are grounded in psychometric intelligence. The other is socioemotional in nature and have answers that change constantly (like what's your favorite color or what you should get someone for Christmas).

The following techniques are usually called ''problem-solving strategies''
* Abstraction: solving the problem in a model of the system before applying it to the real system
* Analogy: using a solution that solves an analogous problem
* Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
* Critical thinking
* Divide and conquer: breaking down a large, complex problem into smaller, solvable problems
* Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
* Lateral thinking: approaching solutions indirectly and creatively
* Means-ends analysis: choosing an action at each step to move closer to the goal
* Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new
* Morphological analysis: assessing the output and interactions of an entire system
* Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
* Reduction: transforming the problem into another problem for which solutions exist
* Research: employing existing ideas or adapting existing solutions to similar problems
* Root cause analysis: identifying the cause of a problem
* Trial-and-error: testing possible solutions until the right one is found

== Multiple forms of representation ==
=== Concrete models===
====Base ten blocks====
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

====Pattern blocks====
[[File:Wooden pattern blocks dodecagon.JPG|thumb|One of the ways of making a dodecagon with pattern blocks]]
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.

Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.

Pattern blocks can also serve to provide students with an understanding of fractions. Because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.

Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

====Unifix® Cubes====
[[File:Linking cm cubes 2.JPG|thumb|Interlocking centimeter cubes]]
Unifix® Cubes are interlocking cubes that are just under 2 centimeters on each side. The cubes connect with each other from one side. Once connected, Unifix® Cubes can be turned to form a vertical Unifix® "tower," or horizontally to form a Unifix® "train".

Other interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities.

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence,
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue..
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.

Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume.

====Tiles====
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).

Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.

These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

====Number lines====
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

====Cuisenaire rods====
[[File:Cuisenaire ten.JPG|thumb|Cuisenaire rods used to illustrate the factors of ten]]
'''Cuisenaire rods''' are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods ''réglettes''.

According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."

;Use in mathematics teaching
The rods are used in teaching a variety of mathematical ideas, and with a wide age range of learners. Topics they are used for include:
* Counting, sequences, patterns and algebraic reasoning
* Addition and subtraction (additive reasoning)
* Multiplication and division (multiplicative reasoning)
* Fractions, ratio and proportion
* Modular arithmetic leading to group theory

=== Pictures ===

=== Diagrams ===
A '''diagram''' is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram.

;Gallery of diagram types

There are at least the following types of diagrams:

* [[Logic]]al or conceptual diagrams, which take a collection of items and relationships between them, and express them by giving each item a 2D position, while the relationships are expressed as connections between the items or overlaps between the items, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Tree Example.png|[[Tree structure|tree diagram]]
File:Neural network.svg|[[Network diagram]]
File:LampFlowchart.svg|[[Flowchart]]
File:Set intersection.svg|[[Venn diagram]]
File:Alphagraphen.png|[[Existential graph]]
</gallery>
|}

* Quantitative diagrams, which display a relationship between two variables that take either [[Discrete mathematics|discrete]] or a [[continuous function|continuous]] range of values; for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Histogram example.svg|[[Histogram]]
File:Graphtestone.svg|[[Bar graph]]
File:Zusammensetzung Shampoo.svg|[[Pie chart]]
File:Hyperbolic Cosine.svg|[[Function graph]]
File:R-car stopping distances 1920.svg|[[Scatter plot]]
File:Hanger Diagram.png|Hanger diagram.
</gallery>
|}

* [[Schematic]]s and other types of diagrams, for example:

{|style="width:100%"
|style="vertical-align:top; width:25px;"|
|style="vertical-align:top;"|
<gallery perrow="5" widths="80" heights="80">
File:Train schedule of Sanin Line, Japan, 1949-09-15, part.png|[[Time–distance diagram]]
File:Gear pump exploded.png|[[Exploded view]]
File:US 2000 census population density map by state.svg|[[Population density|Population density map]]
File:Pioneer plaque.svg|[[Pioneer plaque]]
File:Automotive diagrams 01 En.png|[[Three-dimensional|Three-dimensional diagram]]
</gallery>
|}

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Mathematical_logic Mathematical logic, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Problem_solving Problem solving, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Manipulative_(mathematics_education) Manipulative (mathematics education), Wikipedia] under a CC BY-Sa license
* [https://en.wikipedia.org/wiki/Cuisenaire_rods Cuisenaire rods, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Diagram Diagram, Wikipedia] under a CC BY-SA license

Problem Solving Introduction

2022-02-05T03:23:46Z

Khanh: Created page with "== Licensing == Content obtained and/or adapted from: * [https://en.wikipedia.org/wiki/Base_ten_block Base ten block, Wikipedia] under a CC BY-SA license https://en.wikipedi..."

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Base_ten_block Base ten block, Wikipedia] under a CC BY-SA license

https://en.wikipedia.org/wiki/Mathematical_logic

https://en.wikipedia.org/wiki/Multiple_representations_(mathematics_education)

https://en.wikipedia.org/wiki/Cuisenaire_rods

https://en.wikipedia.org/wiki/Diagram

https://en.wikipedia.org/wiki/Manipulative_(mathematics_education)

https://en.wikipedia.org/wiki/Number_line

https://en.wikipedia.org/wiki/Problem_solving

Sets:Operations

2022-02-05T02:27:40Z

Khanh:

In mathematics, '''the algebra of sets''', not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.

Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being ''union'', the meet operator being ''intersection'', the complement operator being ''set complement'', the bottom being <math>\varnothing</math> and the top being the universe set under consideration.

==Fundamentals==

The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".

It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.

==The fundamental properties of set algebra==

The binary operations of set union (<math>\cup</math>) and intersection (<math>\cap</math>) satisfy many identities. Several of these identities or "laws" have well established names.

:Commutative property:
::*<math>A \cup B = B \cup A</math>
::*<math>A \cap B = B \cap A</math>
:Associative property:
::*<math>(A \cup B) \cup C = A \cup (B \cup C)</math>
::*<math>(A \cap B) \cap C = A \cap (B \cap C)</math>
:Distributive property:
::*<math>A \cup (B \cap C) = (A \cup B) \cap (A \cup C)</math>
::*<math>A \cap (B \cup C) = (A \cap B) \cup (A \cap C)</math>

The union and intersection of sets may be seen as analogous to the addition and multiplication of numbers. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection ''distributes'' over union. However, unlike addition and multiplication, union also distributes over intersection.

Two additional pairs of properties involve the special sets called the empty set Ø and the universe set <math>U</math>; together with the complement operator (<math>A^C</math> denotes the complement of <math>A</math>. This can also be written as <math>A'</math>, read as A prime). The empty set has no members, and the universe set has all possible members (in a particular context).

:Identity :
::*<math>A \cup \varnothing = A</math>
::*<math>A \cap U = A</math>
:Complement :
::*<math>A \cup A^C = U</math>
::*<math>A \cap A^C = \varnothing</math>

The identity expressions (together with the commutative expressions) say that, just like 0 and 1 for addition and multiplication, Ø and '''U''' are the identity elements for union and intersection, respectively.

Unlike addition and multiplication, union and intersection do not have inverse elements. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation of set complementation.

The preceding five pairs of formulae—the commutative, associative, distributive, identity and complement formulae—encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.

Note that if the complement formulae are weakened to the rule <math> (A^C)^C = A </math>, then this is exactly the algebra of propositional linear logic.

==The principle of duality==
Each of the identities stated above is one of a pair of identities such that each can be transformed into the other by interchanging ∪ and ∩, and also Ø and '''U'''.

These are examples of an extremely important and powerful property of set algebra, namely, the '''principle of duality''' for sets, which asserts that for any true statement about sets, the '''dual''' statement obtained by interchanging unions and intersections, interchanging '''U''' and Ø and reversing inclusions is also true. A statement is said to be '''self-dual''' if it is equal to its own dual.

== Some additional laws for unions and intersections ==

The following proposition states six more important laws of set algebra, involving unions and intersections.

'''PROPOSITION 3''': For any subsets ''A'' and ''B'' of a universe set '''U''', the following identities hold:
:idempotent laws:
::*<math>A \cup A = A</math>
::*<math>A \cap A = A</math>
:domination laws:
::*<math>A \cup U = U</math>
::*<math>A \cap \varnothing = \varnothing</math>
:absorption laws:
::*<math>A \cup (A \cap B) = A</math>
::*<math>A \cap (A \cup B) = A</math>

As noted above, each of the laws stated in proposition 3 can be derived from the five fundamental pairs of laws stated above. As an illustration, a proof is given below for the idempotent law for union.

''Proof:''
{|
|-
|<math>A \cup A</math>
|<math>=(A \cup A) \cap U</math>
|by the identity law of intersection
|-
|
|<math>=(A \cup A) \cap (A \cup A^C)</math>
|by the complement law for union
|-
|
|<math>=A \cup (A \cap A^C)</math>
|by the distributive law of union over intersection
|-
|
|<math>=A \cup \varnothing</math>
|by the complement law for intersection
|-
|
|<math>=A</math>
|by the identity law for union
|}
The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law for union, namely the idempotent law for intersection.

''Proof:''
{|
|-
|<math>A \cap A</math>
|<math>=(A \cap A) \cup \varnothing</math>
|by the identity law for union
|-
|
|<math>=(A \cap A) \cup (A \cap A^C)</math>
|by the complement law for intersection
|-
|
|<math>=A \cap (A \cup A^C)</math>
|by the distributive law of intersection over union
|-
|
|<math>=A \cap U</math>
|by the complement law for union
|-
|
|<math>=A</math>
|by the identity law for intersection
|}

Intersection can be expressed in terms of set difference :

<math>A \cap B = A \setminus (A \setminus B) </math>

== Some additional laws for complements ==

The following proposition states five more important laws of set algebra, involving complements.

'''PROPOSITION 4''': Let ''A'' and ''B'' be subsets of a universe '''U''', then:
:De Morgan's laws:
::*<math>(A \cup B)^C = A^C \cap B^C</math>
::*<math>(A \cap B)^C = A^C \cup B^C</math>
:double complement or involution law:
::*<math>{(A^{C})}^{C} = A</math>
:complement laws for the universe set and the empty set:
::*<math>\varnothing^C = U</math>
::*<math>U^C = \varnothing</math>

Notice that the double complement law is self-dual.

The next proposition, which is also self-dual, says that the complement of a set is the only set that satisfies the complement laws. In other words, complementation is characterized by the complement laws.

'''PROPOSITION 5''': Let ''A'' and ''B'' be subsets of a universe '''U''', then:
:uniqueness of complements:
::*If <math>A \cup B = U</math>, and <math>A \cap B = \varnothing</math>, then <math>B = A^C</math>

==The algebra of inclusion==

The following proposition says that inclusion, that is the binary relation of one set being a subset of another, is a partial order.

'''PROPOSITION 6''': If ''A'', ''B'' and ''C'' are sets then the following hold:

:reflexivity:
::*<math>A \subseteq A</math>

:antisymmetry:
::*<math>A \subseteq B</math> and <math>B \subseteq A</math> if and only if <math>A = B</math>

:transitivity:
::*If <math>A \subseteq B</math> and <math>B \subseteq C</math>, then <math>A \subseteq C</math>

The following proposition says that for any set ''S'', the power set of ''S'', ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra.

'''PROPOSITION 7''': If ''A'', ''B'' and ''C'' are subsets of a set ''S'' then the following hold:

:existence of a least element and a greatest element:
::*<math> \varnothing \subseteq A \subseteq S</math>

:existence of joins:
::*<math>A \subseteq A \cup B</math>
::*If <math>A \subseteq C</math> and <math>B \subseteq C</math>, then <math>A \cup B \subseteq C</math>

:existence of meets:
::*<math>A \cap B \subseteq A</math>
::*If <math>C \subseteq A</math> and <math>C \subseteq B</math>, then <math>C \subseteq A \cap B</math>

The following proposition says that the statement <math>A \subseteq B</math> is equivalent to various other statements involving unions, intersections and complements.

'''PROPOSITION 8''': For any two sets ''A'' and ''B'', the following are equivalent:
:*<math>A \subseteq B</math>
:*<math>A \cap B = A</math>
:*<math>A \cup B = B</math>
:*<math>A \setminus B = \varnothing</math>
:*<math>B^C \subseteq A^C</math>

The above proposition shows that the relation of set inclusion can be characterized by either of the operations of set union or set intersection, which means that the notion of set inclusion is axiomatically superfluous.

== The algebra of relative complements ==

The following proposition lists several identities concerning relative complements and set-theoretic differences.

'''PROPOSITION 9''': For any universe '''U''' and subsets ''A'', ''B'', and ''C'' of '''U''', the following identities hold:

:*<math>C \setminus (A \cap B) = (C \setminus A) \cup (C \setminus B)</math>
:*<math>C \setminus (A \cup B) = (C \setminus A) \cap (C \setminus B)</math>
:*<math>C \setminus (B \setminus A) = (A \cap C)\cup(C \setminus B)</math>
:*<math>(B \setminus A) \cap C = (B \cap C) \setminus A = B \cap (C \setminus A)</math>
:*<math>(B \setminus A) \cup C = (B \cup C) \setminus (A \setminus C)</math>
:*<math>(B \setminus A) \setminus C = B \setminus (A \cup C)</math>
:*<math>A \setminus A = \varnothing</math>
:*<math>\varnothing \setminus A = \varnothing</math>
:*<math>A \setminus \varnothing = A</math>
:*<math>B \setminus A = A^C \cap B</math>
:*<math>(B \setminus A)^C = A \cup B^C</math>
:*<math>U \setminus A = A^C</math>
:*<math>A \setminus U = \varnothing</math>

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 101-115

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Algebra_of_sets Algebra of sets, Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T20:03:30Z

Khanh: /* Power sets */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {4, 2, 1, 3}}}
:{{math|1=''B'' = {blue, white, red}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{1, 2, 3, ..., 1000}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{0, 1, 2, 3, 4, ...}}},
and the set of all integers is
:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {1, 2, 3, 4}}}, {{math|1=''B'' = {blue, white, red}}}, and ''F'' = {''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or { } or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {''x''}, where ''x'' is the element.
The set {''x''} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {1, 3} ⊂ {1, 2, 3, 4}.
* {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into 2''n'' zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, <math>-\tfrac{7}{4}</math> ∈ '''Q''' and <math>5 = \tfrac{5}{1}</math> ∈ '''Q''';
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: '''C''' = {''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted |''S''|, is the number of members of {{math|''S''}}. For example, if ''B'' = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted, so

|{blue, white, red, blue, white}| = 3, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{1, 2, 3}}} is

{∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or 2''S''.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has 2''n'' elements. For example, {{math|{1, 2, 3}}} has three elements, and its power set has <math> 2^3 = 8 </math> elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:57:52Z

Khanh: /* Cardinality */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {4, 2, 1, 3}}}
:{{math|1=''B'' = {blue, white, red}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{1, 2, 3, ..., 1000}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{0, 1, 2, 3, 4, ...}}},
and the set of all integers is
:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {1, 2, 3, 4}}}, {{math|1=''B'' = {blue, white, red}}}, and ''F'' = {''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or { } or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {''x''}, where ''x'' is the element.
The set {''x''} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {1, 3} ⊂ {1, 2, 3, 4}.
* {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into 2''n'' zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, <math>-\tfrac{7}{4}</math> ∈ '''Q''' and <math>5 = \tfrac{5}{1}</math> ∈ '''Q''';
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: '''C''' = {''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted |''S''|, is the number of members of {{math|''S''}}. For example, if ''B'' = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted, so

|{blue, white, red, blue, white}| = 3, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:53:30Z

Khanh: /* Special sets of numbers in mathematics */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {4, 2, 1, 3}}}
:{{math|1=''B'' = {blue, white, red}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{1, 2, 3, ..., 1000}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{0, 1, 2, 3, 4, ...}}},
and the set of all integers is
:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {1, 2, 3, 4}}}, {{math|1=''B'' = {blue, white, red}}}, and ''F'' = {''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or { } or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {''x''}, where ''x'' is the element.
The set {''x''} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {1, 3} ⊂ {1, 2, 3, 4}.
* {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into 2''n'' zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, <math>-\tfrac{7}{4}</math> ∈ '''Q''' and <math>5 = \tfrac{5}{1}</math> ∈ '''Q''';
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: '''C''' = {''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:50:16Z

Khanh: /* Euler and Venn diagrams */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {4, 2, 1, 3}}}
:{{math|1=''B'' = {blue, white, red}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{1, 2, 3, ..., 1000}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{0, 1, 2, 3, 4, ...}}},
and the set of all integers is
:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {1, 2, 3, 4}}}, {{math|1=''B'' = {blue, white, red}}}, and ''F'' = {''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or { } or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {''x''}, where ''x'' is the element.
The set {''x''} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {1, 3} ⊂ {1, 2, 3, 4}.
* {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into 2''n'' zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, {{math|−{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:49:09Z

Khanh: /* Subsets */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {4, 2, 1, 3}}}
:{{math|1=''B'' = {blue, white, red}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{1, 2, 3, ..., 1000}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{0, 1, 2, 3, 4, ...}}},
and the set of all integers is
:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {1, 2, 3, 4}}}, {{math|1=''B'' = {blue, white, red}}}, and ''F'' = {''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or { } or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {''x''}, where ''x'' is the element.
The set {''x''} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {1, 3} ⊂ {1, 2, 3, 4}.
* {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into {{math|2{{sup|''n''}}}} zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, {{math|−{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:43:38Z

Khanh: /* Singleton sets */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {4, 2, 1, 3}}}
:{{math|1=''B'' = {blue, white, red}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{1, 2, 3, ..., 1000}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{0, 1, 2, 3, 4, ...}}},
and the set of all integers is
:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {1, 2, 3, 4}}}, {{math|1=''B'' = {blue, white, red}}}, and ''F'' = {''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or { } or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {''x''}, where ''x'' is the element.
The set {''x''} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {{mset|1, 3}} ⊂ {{mset|1, 2, 3, 4}}.
* {{mset|1, 2, 3, 4}} ⊆ {{mset|1, 2, 3, 4}}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into {{math|2{{sup|''n''}}}} zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, {{math|−{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:40:29Z

Khanh: /* The empty set */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {4, 2, 1, 3}}}
:{{math|1=''B'' = {blue, white, red}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{1, 2, 3, ..., 1000}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{0, 1, 2, 3, 4, ...}}},
and the set of all integers is
:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {1, 2, 3, 4}}}, {{math|1=''B'' = {blue, white, red}}}, and ''F'' = {''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or { } or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {{mset|''x''}}, where ''x'' is the element.
The set {{mset|''x''}} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {{mset|1, 3}} ⊂ {{mset|1, 2, 3, 4}}.
* {{mset|1, 2, 3, 4}} ⊆ {{mset|1, 2, 3, 4}}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into {{math|2{{sup|''n''}}}} zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, {{math|−{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:39:18Z

Khanh: /* Membership */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {4, 2, 1, 3}}}
:{{math|1=''B'' = {blue, white, red}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{1, 2, 3, ..., 1000}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{0, 1, 2, 3, 4, ...}}},
and the set of all integers is
:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {1, 2, 3, 4}}}, {{math|1=''B'' = {blue, white, red}}}, and ''F'' = {''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or {{mset| }} or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {{mset|''x''}}, where ''x'' is the element.
The set {{mset|''x''}} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {{mset|1, 3}} ⊂ {{mset|1, 2, 3, 4}}.
* {{mset|1, 2, 3, 4}} ⊆ {{mset|1, 2, 3, 4}}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into {{math|2{{sup|''n''}}}} zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, {{math|−{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:32:53Z

Khanh: /* How sets are defined and set notation */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {4, 2, 1, 3}}}
:{{math|1=''B'' = {blue, white, red}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{1, 2, 3, ..., 1000}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{0, 1, 2, 3, 4, ...}}},
and the set of all integers is
:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {{mset|1, 2, 3, 4}}}}, {{math|1=''B'' = {{mset|blue, white, red}}}}, and {{math|1=''F'' = {{mset|''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19}}}},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or {{mset| }} or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {{mset|''x''}}, where ''x'' is the element.
The set {{mset|''x''}} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {{mset|1, 3}} ⊂ {{mset|1, 2, 3, 4}}.
* {{mset|1, 2, 3, 4}} ⊆ {{mset|1, 2, 3, 4}}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into {{math|2{{sup|''n''}}}} zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, {{math|−{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:30:08Z

Khanh: /* Naïve set theory */

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {{mset|4, 2, 1, 3}}}}
:{{math|1=''B'' = {{mset|blue, white, red}}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{{mset|2, 4, 6}}}} and {{math|{{mset|4, 6, 4, 2}}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{{mset|1, 2, 3, ..., 1000}}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{{mset|0, 1, 2, 3, 4, ...}}}},
and the set of all integers is
:{{math|{{mset|..., −3, −2, −1, 0, 1, 2, 3, ...}}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {{mset|1, 2, 3, 4}}}}, {{math|1=''B'' = {{mset|blue, white, red}}}}, and {{math|1=''F'' = {{mset|''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19}}}},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or {{mset| }} or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {{mset|''x''}}, where ''x'' is the element.
The set {{mset|''x''}} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {{mset|1, 3}} ⊂ {{mset|1, 2, 3, 4}}.
* {{mset|1, 2, 3, 4}} ⊆ {{mset|1, 2, 3, 4}}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into {{math|2{{sup|''n''}}}} zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, {{math|−{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license

Sets:Definitions

2022-02-04T19:22:20Z

Khanh:

[[File:Example of a set.svg|thumb|A set of polygons in an Euler diagram]]

In mathematics, a '''set''' is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

==Origin==
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''.
[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]]
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':

::A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a ''class'':

::When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.

===Naïve set theory===
The foremost property of a set is that it can have elements, also called ''members''. Two sets are equal when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''extensionality of sets''.

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {{mset|''x'' | ''x'' is a set and ''x'' ∉ ''x''}}, cannot exist.
* Cantor's paradox shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any ''well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.

==How sets are defined and set notation==
Mathematical texts commonly denote sets by capital letters in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}. A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
:{{math|1=''A'' = {{mset|4, 2, 1, 3}}}}
:{{math|1=''B'' = {{mset|blue, white, red}}}}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{{mset|2, 4, 6}}}} and {{math|{{mset|4, 6, 4, 2}}}} represent the same set.

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
:{{math|{{mset|1, 2, 3, ..., 1000}}}}.

====Infinite sets in roster notation====
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
:{{math|{{mset|0, 1, 2, 3, 4, ...}}}},
and the set of all integers is
:{{math|{{mset|..., −3, −2, −1, 0, 1, 2, 3, ...}}}}.

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
:Let {{mvar|A}} be the set whose members are the first four positive integers.
:Let {{mvar|B}} be the set of colors of the French flag.

Such a definition is called a ''semantic description''.

===Set-builder notation===

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set {{mvar|F}} can be defined as follows:

:{{mvar|F}} <math> = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.

===Classifying methods of definition===
Philosophy uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''. Such definitions are also called ''enumerative''.
*An ''ostensive definition'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".

For example, with respect to the sets {{math|1=''A'' = {{mset|1, 2, 3, 4}}}}, {{math|1=''B'' = {{mset|blue, white, red}}}}, and {{math|1=''F'' = {{mset|''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19}}}},
:{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
:{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

==The empty set==
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or {{mset| }} or {{math|ϕ}} (or {{mvar|ϕ}}).

==Singleton sets==
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {{mset|''x''}}, where ''x'' is the element.
The set {{mset|''x''}} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'', or ''B'' ⊇ ''A''. The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The relationship between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset), while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {{mset|1, 3}} ⊂ {{mset|1, 2, 3, 4}}.
* {{mset|1, 2, 3, 4}} ⊆ {{mset|1, 2, 3, 4}}.

The empty set is a subset of every set, and every set is a subset of itself:
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''. ''B'' is a superset of ''A''.]]
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into {{math|2{{sup|''n''}}}} zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\bold Z</math>) or blackboard bold (e.g. <math>\mathbb Z</math>) typeface. These include
* <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
* <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, {{math|−{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};
* <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''function'' (or ''mapping'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of relation, one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''infinite''. For example, the set <math>\N</math> of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have ''infinite cardinality''.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of <math>\N</math> are called ''countable sets''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''uncountable sets''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

===The Continuum Hypothesis===
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

==Partitions==

A partition of a set ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is ''S''.

== Basic operations ==

There are several fundamental operations for constructing new sets from given sets.

=== Unions ===
[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
Two sets can be joined: the ''union'' of {{math|''A''}} and {{math|''B''}}, denoted by {{math|''A'' ∪ ''B''}}, is the set of all things that are members of ''A'' or of ''B'' or of both.

Examples:
* {{math|1={1, 2} ∪ {1, 2} = {1, 2}.}}
* {{math|1={1, 2} ∪ {2, 3} = {1, 2, 3}.}}
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.}}

'''Some basic properties of unions:'''
* {{math|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
* {{math|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
* {{math|1=''A'' ⊆ (''A'' ∪ ''B'').}}
* {{math|1=''A'' ∪ ''A'' = ''A''.}}
* {{math|1=''A'' ∪ ∅ = ''A''.}}
* {{math|''A'' ⊆ ''B''}} if and only if {{math|1=''A'' ∪ ''B'' = ''B''.}}

=== Intersections ===
A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by ''A'' ∩ ''B'', is the set of all things that are members of both ''A'' and ''B''. If ''A'' ∩ ''B'' = ∅, then ''A'' and ''B'' are said to be ''disjoint''.
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted ''A'' ∩ ''B''.</div>]]

Examples:
* {1, 2} ∩ {1, 2} = {1, 2}.
* {1, 2} ∩ {2, 3} = {2}.
* {1, 2} ∩ {3, 4} = ∅.

'''Some basic properties of intersections:'''
* ''A'' ∩ ''B'' = ''B'' ∩ ''A''.
* ''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.
* ''A'' ∩ ''B'' ⊆ ''A''.
* ''A'' ∩ ''A'' = ''A''.
* ''A'' ∩ ∅ = ∅.
* ''A'' ⊆ ''B'' if and only if ''A'' ∩ ''B'' = ''A''.

=== Complements ===
[[File:Venn0100.svg|thumb|<div class="center">The '''relative complement''' of ''B'' in ''A''</div>]]
[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by ''A'' \ ''B'' (or ''A'' − ''B''), is the set of all elements that are members of ''A,'' but not members of ''B''. It is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set ''U''. In such cases, ''U'' \ ''A'' is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′ or Ac.
* ''A''′ = ''U'' \ ''A''

Examples:
* {1, 2} \ {1, 2} = ∅.
* {1, 2, 3, 4} \ {1, 3} = {2, 4}.
* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.

Some basic properties of complements include the following:
* ''A'' \ ''B'' ≠ ''B'' \ ''A'' for ''A'' ≠ ''B''.
* ''A'' ∪ ''A''′ = ''U''.
* ''A'' ∩ ''A''′ = ∅.
* (''A''′)′ = ''A''.
* ∅ \ ''A'' = ∅.
* ''A'' \ ∅ = ''A''.
* ''A'' \ ''A'' = ∅.
* ''A'' \ ''U'' = ∅.
* ''A'' \ ''A''′ = ''A'' and ''A''′ \ ''A'' = ''A''′.
* ''U''′ = ∅ and ∅′ = ''U''.
* ''A'' \ ''B'' = ''A'' ∩ ''B''′.
* if ''A'' ⊆ ''B'' then ''A'' \ ''B'' = ∅.

An extension of the complement is the symmetric difference, defined for sets ''A'', ''B'' as
:<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

=== Cartesian product ===
A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B,'' is the set of all ordered pairs (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.

Examples:
* {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
* {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
* {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:
* ''A'' × ∅ = ∅.
* ''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').
* (''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').
Let ''A'' and ''B'' be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
* |''A'' × ''B''| = |''B'' × ''A''| = |''A''| × |''B''|.

==Applications==
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain {{math|''A''}} to a codomain {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set ''S'' = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set ''B'' = {(scissors,paper), (paper,rock), (rock,scissors)}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs (''x'', ''x''2), where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a function. In functional notation, this relation can be written as ''F''(''x'') = ''x''2.

==Principle of inclusion and exclusion==
[[Image:A union B.svg|thumb|The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.]]

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
:<math> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle can be used to find the cardinality of any finite union of sets:
:<math>\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>

== De Morgan's laws ==
Augustus De Morgan stated two laws about sets.

If {{mvar|A}} and {{mvar|B}} are any two sets then,
* '''(''A'' ∪ ''B'')′ = ''A''′ ∩ ''B''′'''
The complement of {{mvar|A}} union {{mvar|B}} equals the complement of {{mvar|A}} intersected with the complement of {{mvar|B}}.
* '''(''A'' ∩ ''B'')′ = ''A''′ ∪ ''B''′'''
The complement of {{mvar|A}} intersected with {{mvar|B}} is equal to the complement of {{mvar|A}} union to the complement of {{mvar|B}}.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 93-100

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Set_(mathematics) Set (mathematics), Wikipedia] under a CC BY-SA license