MAT3013

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

• Ethan D. Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics, 2nd ed, Springer (2011). https://link-springer-com.libweb.lib.utsa.edu/book/10.1007%2F978-1-4419-7127-2

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	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		Motivation for rigorous mathematics from a historical perspective An understanding of where and why this course is going
2.0	1.2-3	Logical Operators		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		Gain the prerequisites for writing and evaluating proofs.
3.0	1.4-6	Methods for Proofs	Propositions Logical Operators	Start proving elementary results.
4.0	2.1-3	Basic Concepts of Set Theory		How to start working with sets
4.0	2.1-3	Operations with sets	Basic Concepts of Set Theory	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	2.4-6	Counting Principles	Natural Numbers	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	Cartesian Products	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	Operations with sets Constructions with sets	Gain basic concepts about relations.
8.0	3.4-5	Partial Orders	Equivalence Relations	Familiarize with ordering. Learn how to use graph representations of relations.
8.0	3.4-5	Graphs	Equivalence Relations	Familiarize with ordering. Learn how to use graph representations of relations.
9.0	4.1-2	Functions	Equivalence Relations 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.
10.0	4.3-4	One-to-One	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	Finding inverses
11.0	4.5-6	Images of Subsets	One-to-One Onto Compositional Inverse	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		Catch up and review Midterm 2
13.0	5.1-2	Finite Sets	Functions Operations with sets Constructions with sets	Learn classification of sets by size.
13.0	5.1-2	Infinite Sets	Functions Operations with sets Constructions with sets	Learn classification of sets by size. Generalizing the concept of size to infinite sets
13.0	5.1-2	Equivalent Sets	Functions Operations with sets Constructions with sets
14.0	5.3-5	Uncountable Sets	Finite Sets Infinite Sets Equivalent 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		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 Logical statements	1.1-1.2		Motivation for rigorous mathematics from a historical perspective An understanding of where and why this course is going
2	Informal logic	Statements Relation between statements Valid Arguments Quantifiers	1.1-1.5		Gain the prerequisites for writing and evaluating proofs.	connectives conditionals biconditionals
3	Strategies for proofs	Why we need proofs Direct proofs Proofs by contrapositive and contradiction Cases and If and Only If	2.2-2.4	informal logic	Start proving elementary results.	direct proofs modus ponens proofs by contradiction
4	Writing Mathematics/Set theory I	Basic concepts Operations and constructions with sets	2.6, 3.1-3.3	Writing math Basic concepts of set theory Set operations	How to start working with sets	notation subsets proving sets are equal unions, intersections, complements
5	Set theory II	Family of sets Axioms of set theory	3.4-3.5	Natural numbers	Learn constructive proofs and reasoning. Learn basic axiom of set theory	well ordering principle inclusion-exclusion principle
6	Catch up and review Midterm 1
7	Functions I	Definition of functions Image and inverse image Composition and inverse functions	4.1-4.3	Set theory	Gain basic concepts about functions.
8	Functions II	Injectivity, surjectivity and bijectivity Sets of functions	4.4-4.5			Examples
9	Relations I	Relations Congruence	5.1-5.2	Relations Function sense (precalculus)	Gain basic rigorous knowledge of relations.
10	Relations II	Equivalence relations	4.3-4	Functions 1
11	Finite and infinite sets II	Introduction Properties of natural numbers	6.1-6.2	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	Finite and infinite sets II	Mathematical induction Recursion	6.2-6.3	Sets and functions	Learn classification of sets by size. Generalizing the concept of size to infinite sets
14	Finite and infinite sets III	Cardinality of sets Finite sets and countable sets Cardinality of number systems	6.4 - 6.7	Cardinality 1	Learn properties of countable sets.
15	Catch up and review for final Study days

Topics List C (Proofs and Fundamentals)

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

• UTSA Undergraduate Mathematics Course Descriptions