Difference between revisions of "Statements"
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In mathematics, a statement (also known as a proposition) is a declarative sentence that is either true or false but not both. There must be no ambiguity as to whether the statement is true or false. The statement cannot be an opinion or vague sentence (for example, "chocolate tastes good" or "red is an ugly color") that does not have a set truth value. Some examples of basic statements and their truth values: | In mathematics, a statement (also known as a proposition) is a declarative sentence that is either true or false but not both. There must be no ambiguity as to whether the statement is true or false. The statement cannot be an opinion or vague sentence (for example, "chocolate tastes good" or "red is an ugly color") that does not have a set truth value. Some examples of basic statements and their truth values: | ||
| − | * 2 is an even number (TRUE) | + | * <math> 2 </math> is an even number (TRUE) |
| − | * Negative numbers are less than 0 (TRUE) | + | * Negative numbers are less than <math> 0 </math> (TRUE) |
| − | * 3 + 4 = 6 (FALSE) | + | * <math> 3 + 4 = 6 </math> (FALSE) |
* Quadrilaterals have five sides (FALSE) | * Quadrilaterals have five sides (FALSE) | ||
* For all real numbers <math> n </math>, <math> n^2 \ge 0 </math> (TRUE) | * For all real numbers <math> n </math>, <math> n^2 \ge 0 </math> (TRUE) | ||
| − | * All multiples of 3 are even (FALSE) | + | * All multiples of <math> 3 </math> are even (FALSE) |
===Connectives=== | ===Connectives=== | ||
| − | We can use connectives to create compound statements out of two or more basic statements, or to transform statements. Here are some common connectives and how they are used (let P and Q be statements): | + | We can use connectives to create compound statements out of two or more basic statements, or to transform statements. Here are some common connectives and how they are used (let <math> P </math> and <math> Q </math> be statements): |
| − | * Conjunction: "P and Q", also denoted as "P | + | * Conjunction: "<math> P </math> and <math> Q </math>", also denoted as "<math> P \and Q </math>". A conjunctive statement is only true if both <math> P </math> and <math> Q </math> are true, and false otherwise. |
| − | * Disjunction: " P or Q", also denoted as "P | + | * Disjunction: "<math> P </math> or <math> Q </math>", also denoted as "<math> P \or Q </math>". A disjunctive statement is true if either <math> P </math> or <math> Q </math> are true, and false only when both <math> P </math> and <math> Q </math> are false. |
| − | * Negation: "not P", also denoted as " | + | * Negation: "not <math> P </math>", also denoted as "<math> \neg P </math>". A negation reverses the truth value of the original statement (that is, if <math> P </math> is true then <math> \neg P </math> is false, and vice versa). |
| − | * Implication: "if P, then Q" or "P implies Q", also denoted as " | + | * Implication: "if <math> P </math>, then <math> Q </math>" or "<math> P </math> implies <math> Q </math>", also denoted as "<math> P \rightarrow Q</math>". An implication is true if <math> P </math> and <math> Q </math> are both true, as well as when <math> P </math> is false, regardless of the truth value of <math> Q </math> (that is, <math> Q </math> can be true or false when <math> P </math> is false, and the implication will still be true). An implication is only false if <math> P </math> is true and <math> Q </math> is false. |
| − | * Biconditional: "P if and only if Q", also denoted as " | + | * Biconditional: "<math> P </math> if and only if <math> Q </math>", also denoted as "<math> P \Leftrightarrow Q </math>". A biconditional is only true when both <math> P \Rightarrow Q </math> and <math> Q \Rightarrow P </math>" are true statements; that is, <math> P \Leftrightarrow Q </math> is true when <math> P </math> and <math> Q </math> have the same truth value, and false when <math> P </math> and <math> Q </math> have different truth values. |
| − | Truth table of some basic logical statements given statements P and Q: | + | Truth table of some basic logical statements given statements <math> P </math> and <math> Q </math>: |
{| class="wikitable" style="margin:1em auto 1em auto; text-align:center;" | {| class="wikitable" style="margin:1em auto 1em auto; text-align:center;" | ||
|- | |- | ||
Revision as of 13:03, 27 September 2021
In mathematics, a statement (also known as a proposition) is a declarative sentence that is either true or false but not both. There must be no ambiguity as to whether the statement is true or false. The statement cannot be an opinion or vague sentence (for example, "chocolate tastes good" or "red is an ugly color") that does not have a set truth value. Some examples of basic statements and their truth values:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 } is an even number (TRUE)
- Negative numbers are less than (TRUE)
- (FALSE)
- Quadrilaterals have five sides (FALSE)
- For all real numbers , (TRUE)
- All multiples of are even (FALSE)
Connectives
We can use connectives to create compound statements out of two or more basic statements, or to transform statements. Here are some common connectives and how they are used (let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } be statements):
- Conjunction: "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } ", also denoted as "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \and Q } ". A conjunctive statement is only true if both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } are true, and false otherwise.
- Disjunction: "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } ", also denoted as "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \or Q } ". A disjunctive statement is true if either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } are true, and false only when both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } are false.
- Negation: "not Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } ", also denoted as "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg P } ". A negation reverses the truth value of the original statement (that is, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } is true then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg P } is false, and vice versa).
- Implication: "if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } " or "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } ", also denoted as "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \rightarrow Q} ". An implication is true if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } are both true, as well as when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } is false, regardless of the truth value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } (that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } can be true or false when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } is false, and the implication will still be true). An implication is only false if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } is true and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } is false.
- Biconditional: "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } ", also denoted as "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \Leftrightarrow Q } ". A biconditional is only true when both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \Rightarrow Q } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q \Rightarrow P } " are true statements; that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \Leftrightarrow Q } is true when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } have the same truth value, and false when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } have different truth values.
Truth table of some basic logical statements given statements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } :
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \lor Q} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \Rightarrow Q} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q \Rightarrow P} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \Leftrightarrow Q} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg P } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg Q } | |
|---|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | F | F |
| T | F | F | T | F | T | F | F | T |
| F | T | F | T | T | F | F | T | F |
| F | F | F | F | T | T | T | T | T |
Resources
- Course Textbook, pages 4-25
- Statements and Logical Operators, OpenStax
