Difference between revisions of "MAT3013"

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(→‎Topics List A: Split The topics into their own rows, and created links to their prereqs and pages for each topic.)
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* Historical remarks
 
* Historical remarks
 
* Overview of the course and its goals
 
* Overview of the course and its goals
−
* Ideas of proofs and logic
+
||
−
* Axioms and propositions
+
 
 +
||
 +
* Motivation for rigorous mathematics from a historical perspective
 +
* An understanding of where and why this course is going
 +
|-
 +
|1.0
 +
||
 +
* 1.1
 +
||
 +
* [[Proofs]]
 +
||
 +
 
 +
||
 +
* Motivation for rigorous mathematics from a historical perspective
 +
* An understanding of where and why this course is going
 +
|-
 +
|1.0
 +
||
 +
* 1.1
 +
||
 +
* [[Logic]]
 +
||
 +
 
 +
||
 +
* Motivation for rigorous mathematics from a historical perspective
 +
* An understanding of where and why this course is going
 +
|-
 +
|1.0
 +
||
 +
* 1.1
 +
||
 +
* [[Axioms]]
 +
||
 +
 
 +
||
 +
* Motivation for rigorous mathematics from a historical perspective
 +
* An understanding of where and why this course is going
 +
|-
 +
|1.0
 +
||
 +
* 1.1
 +
||
 +
* [[Propositions]]
 
||
 
||
  
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* 1.2-3
 
* 1.2-3
 
||
 
||
−
* Logical operators
+
* [[Logical Operators]]
−
* Truth values
+
||
−
* Truth tables
+
 
−
* Quantifiers
+
||
 +
* Gain the prerequisites for writing and evaluating proofs.
 +
|-
 +
|2.0
 +
||
 +
* 1.2-3
 +
||
 +
* [[Truth Values]]
 +
||
 +
 
 +
||
 +
* Gain the prerequisites for writing and evaluating proofs.
 +
|-
 +
|2.0
 +
||
 +
* 1.2-3
 +
||
 +
* [[Truth Tables]]
 +
||
 +
 
 +
||
 +
* Gain the prerequisites for writing and evaluating proofs.
 +
|-
 +
|2.0
 +
||
 +
* 1.2-3
 +
||
 +
* [[Quantifiers]]
 
||
 
||
  
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* 1.4-6
 
* 1.4-6
 
||
 
||
−
* Methods for proofs
+
* [[Methods for Proofs]]
 
||
 
||
 
* Propositional logic
 
* Propositional logic
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* 2.1-3
 
* 2.1-3
 
||
 
||
−
* Basic concepts
+
* [[Basic Concepts of Set Theory]]
−
* Operations and constructions with sets
+
||
 +
 
 +
||
 +
* How to start working with sets
 +
|-
 +
|4.0
 +
||
 +
* 2.1-3
 +
||
 +
* [[Operations with sets]]
 
||
 
||
−
* Basic concepts of set theory
+
* [[Basic Concepts of Set Theory]]
 
||
 
||
 
* How to start working with sets
 
* How to start working with sets
 +
|-
 +
|4.0
 +
||
 +
* 2.1-3
 +
||
 +
* [[Constructions with sets]]
 +
||
 +
* [[Basic Concepts of Set Theory]]
 +
||
 +
* How to start working with sets
 +
|-
 +
|5.0
 +
||
 +
* 2.4-6
 +
||
 +
* [[Mathematical Induction]]
 +
||
 +
* [[Natural Numbers]]
 +
||
 +
* Learn constructive proofs and reasoning.
 +
* Learn basic counting principles of discrete mathematics.
 
|-
 
|-
 
|5.0
 
|5.0
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* 2.4-6
 
* 2.4-6
 
||
 
||
−
* Mathematical induction
+
* [[Counting Principles]]
−
* Counting principles
 
 
||
 
||
−
* Natural numbers
+
* [[Natural Numbers]]
 
||
 
||
 
* Learn constructive proofs and reasoning.
 
* Learn constructive proofs and reasoning.
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* 3.1-3
 
* 3.1-3
 
||
 
||
−
* Cartesian products and their subsets
+
* [[Cartesian Products]]
−
* Equivalence relations
+
* [[Cartesian Products Subsets]]
 +
* [[Equivalence Relations]]
 +
||
 +
* [[Operations with sets]]
 +
* [[Constructions with sets]]
 +
||
 +
* Gain basic concepts about relations.
 +
|-
 +
|7.0
 +
||
 +
* 3.1-3
 +
||
 +
* [[Cartesian Products Subsets]]
 +
||
 +
* [[Operations with sets]]
 +
* [[Constructions with sets]]
 +
||
 +
* Gain basic concepts about relations.
 +
|-
 +
|7.0
 +
||
 +
* 3.1-3
 +
||
 +
* [[Equivalence Relations]]
 
||
 
||
−
* Set theory
+
* [[Operations with sets]]
 +
* [[Constructions with sets]]
 
||
 
||
 
* Gain basic concepts about relations.
 
* Gain basic concepts about relations.
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* 3.4-5
 
* 3.4-5
 
||
 
||
−
* Partial orders
+
* [[Partial Orders]]
−
* Graphs
 
 
||
 
||
−
* Relations 1
+
* [[Equivalence Relations]]
 +
||
 +
* Familiarize with ordering.
 +
* Learn how to use graph representations of relations.
 +
|-
 +
|8.0
 +
||
 +
* 3.4-5
 +
||
 +
* [[Graphs]]
 +
||
 +
* [[Equivalence Relations]]
 
||
 
||
 
* Familiarize with ordering.
 
* Familiarize with ordering.
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* 4.1-2
 
* 4.1-2
 
||
 
||
−
* Functions
+
* [[Functions]]
−
* Constructions with functions
 
 
||
 
||
−
* Relations
+
* [[Equivalence Relations]]
−
* Function sense (precalculus)
+
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
 +
||
 +
* Gain basic rigorous knowledge of functions.
 +
|-
 +
|9.0
 +
||
 +
* 4.1-2
 +
||
 +
* [[Constructions With Functions]]
 +
||
 +
* [[Equivalence Relations]]
 +
* [[Functions and Their Graphs]] (MAT 1093: Precalculus)
 
||
 
||
 
* Gain basic rigorous knowledge of functions.
 
* Gain basic rigorous knowledge of functions.
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* 4.3-4
 
* 4.3-4
 
||
 
||
−
* One-to-one
+
* [[One-to-One]]
−
* Onto
 
−
* Compositional inverse
 
 
||
 
||
−
* Functions 1
+
* [[Functions]]
 +
* [[Constructions With Functions]]
 +
||
 +
* Determine whether a function is one-to-one with proofs.
 +
|-
 +
|10.0
 +
||
 +
* 4.3-4
 +
||
 +
* [[Onto]]
 +
||
 +
* [[Functions]]
 +
* [[Constructions With Functions]]
 +
||
 +
* Determine whether a function onto with proofs.
 +
|-
 +
|10.0
 +
||
 +
* 4.3-4
 +
||
 +
* [[Compositional Inverse]]
 +
||
 +
* [[Functions]]
 +
* [[Constructions With Functions]]
 
||
 
||
−
* Determine whether a function is one-to-one of onto, with proofs.
 
 
* Finding inverses
 
* Finding inverses
 
|-
 
|-
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* 4.5-6
 
* 4.5-6
 
||
 
||
−
* Images of subsets
+
* [[Images of Subsets]]
−
* Preimages of subsets
 
−
* Sequences
 
 
||
 
||
−
* Functions 2
+
* [[One-to-One]]
 +
* [[Onto]]
 +
* [[Compositional Inverse]]
 
||
 
||
−
* Find images and preimages of subsets under functions, with proofs.
+
* Find images of subsets under functions, with proofs.
 +
|-
 +
|11.0
 +
||
 +
* 4.5-6
 +
||
 +
* [[Preimages of subsets]]
 +
||
 +
* [[One-to-One]]
 +
* [[Onto]]
 +
* [[Compositional Inverse]]
 +
||
 +
* Find preimages of subsets under functions, with proofs.
 +
|-
 +
|11.0
 +
||
 +
* 4.5-6
 +
||
 +
* [[Sequences]]
 +
||
 +
* [[One-to-One]]
 +
* [[Onto]]
 +
* [[Compositional Inverse]]
 +
||
 +
 
 
|-
 
|-
 
|12.0
 
|12.0
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* 5.1-2
 
* 5.1-2
 
||
 
||
−
* Finite and infinite sets
+
* [[Finite Sets]]
−
* Equivalent sets
+
||
 +
* [[Functions]]
 +
* [[Operations with sets]]
 +
* [[Constructions with sets]]
 +
||
 +
* Learn classification of sets by size.
 +
|-
 +
|13.0
 +
||
 +
* 5.1-2
 +
||
 +
* [[Infinite Sets]]
 
||
 
||
−
* Sets and functions
+
* [[Functions]]
 +
* [[Operations with sets]]
 +
* [[Constructions with sets]]
 
||
 
||
 
* Learn classification of sets by size.
 
* Learn classification of sets by size.
 
* Generalizing the concept of size to infinite sets
 
* Generalizing the concept of size to infinite sets
 +
|-
 +
|13.0
 +
||
 +
* 5.1-2
 +
||
 +
* [[Equivalent Sets]]
 +
||
 +
* [[Functions]]
 +
* [[Operations with sets]]
 +
* [[Constructions with sets]]
 +
||
 +
 
|-
 
|-
 
|14.0
 
|14.0
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* 5.3-5
 
* 5.3-5
 
||
 
||
−
* Countable and uncountable sets
+
* [[Uncountable Sets]]
 
||
 
||
−
* Cardinality 1
+
* [[Finite Sets]]
 +
* [[Infinite Sets]]
 +
* [[Equivalent Sets]]
 
||
 
||
 
* Learn properties of countable sets.
 
* Learn properties of countable sets.
 +
|-
 +
|14.0
 +
||
 +
* 5.3-5
 +
||
 +
* [[Uncountable Sets]]
 +
||
 +
* [[Finite Sets]]
 +
* [[Infinite Sets]]
 +
* [[Equivalent Sets]]
 +
||
 +
* Learn properties of uncountable sets.
 
|-
 
|-
 
|15.0
 
|15.0

Revision as of 10:32, 23 July 2020

Foundations of Mathematics (3-0) 3 Credit Hours

Course Catalog

MAT 3013. Foundations of Mathematics. (3-0) 3 Credit Hours.

Prerequisite: MAT1214. Development of theoretical tools for rigorous mathematics. Topics may include mathematical logic, propositional and predicate calculus, set theory, functions and relations, cardinal and ordinal numbers, Boolean algebras, and construction of the natural numbers, integers, and rational numbers. Emphasis on theorem proving. (Formerly MAT2243. Credit cannot be earned for MAT3013 and MAT2243.) Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

Description

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs (notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and products and unions of sets, vital to further work in mathematics.

Evaluation

  • No makeup exams are offered.
  • An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade

could be used as the grade for the missed exam.

  • Students will have access to several past exams for practice.

Text

D. Smith, M. Eggen, R. St. Andre, A Transition to Advanced Mathematics (7e), Brooks/Cole

Topics List A

Date Sections Topics Prerequisite Skills Student Learning Outcomes
1.0
  • 1.1
  • Historical remarks
  • Overview of the course and its goals
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
1.0
  • 1.1
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
1.0
  • 1.1
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
1.0
  • 1.1
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
1.0
  • 1.1
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
2.0
  • 1.2-3
  • Gain the prerequisites for writing and evaluating proofs.
2.0
  • 1.2-3
  • Gain the prerequisites for writing and evaluating proofs.
2.0
  • 1.2-3
  • Gain the prerequisites for writing and evaluating proofs.
2.0
  • 1.2-3
  • Gain the prerequisites for writing and evaluating proofs.
3.0
  • 1.4-6
  • Propositional logic
  • Start proving elementary results.
4.0
  • 2.1-3
  • How to start working with sets
4.0
  • 2.1-3
  • How to start working with sets
4.0
  • 2.1-3
  • How to start working with sets
5.0
  • 2.4-6
  • Learn constructive proofs and reasoning.
  • Learn basic counting principles of discrete mathematics.
5.0
  • 2.4-6
  • Learn constructive proofs and reasoning.
  • Learn basic counting principles of discrete mathematics.
6.0
  • Catch up and review
  • Midterm 1
7.0
  • 3.1-3
  • Gain basic concepts about relations.
7.0
  • 3.1-3
  • Gain basic concepts about relations.
7.0
  • 3.1-3
  • Gain basic concepts about relations.
8.0
  • 3.4-5
  • Familiarize with ordering.
  • Learn how to use graph representations of relations.
8.0
  • 3.4-5
  • Familiarize with ordering.
  • Learn how to use graph representations of relations.
9.0
  • 4.1-2
  • Gain basic rigorous knowledge of functions.
9.0
  • 4.1-2
  • Gain basic rigorous knowledge of functions.
10.0
  • 4.3-4
  • Determine whether a function is one-to-one with proofs.
10.0
  • 4.3-4
  • Determine whether a function onto with proofs.
10.0
  • 4.3-4
  • Finding inverses
11.0
  • 4.5-6
  • Find images of subsets under functions, with proofs.
11.0
  • 4.5-6
  • Find preimages of subsets under functions, with proofs.
11.0
  • 4.5-6
12.0
  • Catch up and review
  • Midterm 2
13.0
  • 5.1-2
  • Learn classification of sets by size.
13.0
  • 5.1-2
  • Learn classification of sets by size.
  • Generalizing the concept of size to infinite sets
13.0
  • 5.1-2
14.0
  • 5.3-5
  • Learn properties of countable sets.
14.0
  • 5.3-5
  • Learn properties of uncountable sets.
15.0
  • Catch up and review for Final
  • Study Days

Topics List B

Week Session Topics Section Prerequisite skills Learning outcomes Examples
1 Introduction
  • Historical remarks
  • Overview of the course and its goals
  • Ideas of proofs and logic
  • Axioms and propositions
 1.1
  • Motivation for rigorous

mathematics from a historical perspective

  • An understanding of where

and why this course is going

2 Propositional logic
  • Logical operators
  • Truth values
  • Truth tables
  • Quantifiers
 1.2-3 Gain the prerequisites for

writing and evaluating proofs.

  • connectives
  • conditionals
  • biconditionals
3 Proof methods Methods for proofs 1.4-6 Propositional logic Start proving elementary results.
  • direct proofs
  • modus ponens
  • proofs by contradiction
4 Set theory
  • Basic concepts
  • Operations and constructions with sets
 2.1-3 Basic concepts of set theory How to start working with sets
  • notation
  • subsets
  • proving sets are equal
  • unions, intersections, complements
5 Induction and counting
  • Mathematical induction
  • Counting principles
 2.4-6 Natural numbers
  • Learn constructive proofs and reasoning.
  • Learn basic counting principles of discrete mathematics.
  • sums of consecutive powers
  • other induction proofs
  • well ordering principle
  • inclusion-exclusion principle
6
  • Catch up and review
  • Midterm 1
7 Relations 1
  • Cartesian products and their subsets
  • Equivalence relations
 3.1-3 Set theory Gain basic concepts about relations.
  • modular congruence
  • gluing sets
8 Relations 2
  • Partial orders
  • Graphs
 3.4-5 Relations 1
  • Familiarize with ordering.
  • Learn how to use graph representations of relations.
 partial ordering of the power set under inclusion
9 Functions 1
  • Functions
  • Constructions with functions
 4.1-2
  • Relations
  • Function sense (precalculus)
 Gain basic rigorous knowledge of functions.

functional composition

10 Functions 2
  • One-to-one
  • Onto
  • Compositional inverse
 4.3-4 Functions 1
  • Determine whether a function is one-to-one of onto, with proofs.
  • Finding inverses
  • examples with finite sets
  • many precalculus examples
11 Functions 3
  • Images of subsets
  • Preimages of subsets
  • Sequences
 4.5-6 Functions 2 Find images and preimages of subsets under functions, with proofs.
  • examples with finite sets
  • many precalculus examples
12
  • Catch up and review
  • Midterm 2
13 Cardinality 1
  • Finite and infinite sets
  • Equivalent sets
 5.1-2 Sets and functions
  • Learn classification of sets by size.
  • Generalizing the concept of size to infinite sets
14 Cardinality 2 Countable and uncountable sets 5.3-5 Cardinality 1 Learn properties of countable sets.
15
  • Catch up and review for final
  • Study days

See also

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=MAT3013&oldid=307"