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 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 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	Prerequisites	Outcomes	Examples
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	Prerequisites	Outcomes	Examples
4	Writing Mathematics/Set theory I	Basic concepts Operations and constructions with sets	2.6, 3.1-3.3	Prerequisites	Outcomes	Examples
5	Set theory II	Family of sets Axioms of set theory	3.4-3.5	Prerequisites	Outcomes	Examples
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	Prerequisites	Outcomes	Examples
8	Functions II	Injectivity, surjectivity and bijectivity Sets of functions	4.4-4.5	Prerequisites	Outcomes	Examples
9	Relations I	Relations Congruence	5.1-5.2	Prerequisites	Outcomes	Examples
10	Relations II	Equivalence relations	4.3-4	Prerequisites	Outcomes	Examples
11	Finite and infinite sets II	Introduction Properties of natural numbers	6.1-6.2	Prerequisites	Outcomes	Examples
12	Catch up and review Midterm 2
13	Finite and infinite sets II	Mathematical induction Recursion	6.2-6.3	Prerequisites	Outcomes	Examples
14	Finite and infinite sets III	Cardinality of sets Finite sets and countable sets Cardinality of number systems	6.4 - 6.7	Prerequisites	Outcomes	Examples
15	Catch up and review for final Study days

Topics List D (Proofs and Fundamentals) Wiki Format

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
1.	1.1-1.2	Historical remarks	Prerequisites	Motivation for rigorous mathematics from a historical perspective An understanding of where and why this course is going
1.	1.1-1.2	Overview of the course and its goals	Prerequisites	Motivation for rigorous mathematics from a historical perspective An understanding of where and why this course is going
1.	1.1-1.2	Ideas of proofs and logic	Prerequisites	Motivation for rigorous mathematics from a historical perspective An understanding of where and why this course is going
1.	1.1-1.2	Logical statements	Prerequisites	Motivation for rigorous mathematics from a historical perspective An understanding of where and why this course is going
2.	1.1-1.5	Statements	Prerequisites	Outcomes
2.	1.1-1.5	Relation between statements	Prerequisites	Outcomes
2.	1.1-1.5	Valid Arguments	Prerequisites	Outcomes
2.	1.1-1.5	Quantifiers	Prerequisites	Outcomes
3.	2.2-2.4	Why we need proofs	Prerequisites	Outcomes
3.	2.2-2.4	Direct proofs	Prerequisites	Outcomes
3.	2.2-2.4	Proofs by contrapositive and contradiction	Prerequisites	Outcomes
3.	2.2-2.4	Cases and If and Only If	Prerequisites	Outcomes
4.	2.6, 3.1-3.3	Basic concepts	Prerequisites	Outcomes
4.	2.6, 3.1-3.3	Operations and constructions with sets	Prerequisites	Outcomes
5.	3.4-3.5	Family of sets	Prerequisites	Outcomes
5.	3.4-3.5	Axioms of set theory	Prerequisites	Outcomes
6.
7.	4.1-4.3	Definition of functions	Prerequisites	Outcomes
7.	4.1-4.3	Image and inverse image	Prerequisites	Outcomes
7.	4.1-4.3	Composition and inverse functions	Prerequisites	Outcomes
8.	4.4-4.5	Injectivity, surjectivity and bijectivity	Prerequisites	Outcomes
8.	4.4-4.5	Sets of functions	Prerequisites	Outcomes
9.	5.1-5.2	Relations	Prerequisites	Outcomes
9.	5.1-5.2	Congruence	Prerequisites	Outcomes
10.	4.3-4	Equivalence relations	Prerequisites	Outcomes
11.	6.1-6.2	Introduction	Prerequisites	Outcomes
11.	6.1-6.2	Properties of natural numbers	Prerequisites	Outcomes
12.
13.	6.2-6.3	Mathematical induction	Prerequisites	Outcomes
13.	6.2-6.3	Recursion	Prerequisites	Outcomes
14.	6.4 - 6.7	Cardinality of sets	Prerequisites	Outcomes
14.	6.4 - 6.7	Finite sets and countable sets	Prerequisites	Outcomes
14.	6.4 - 6.7	Cardinality of number systems	Prerequisites	Outcomes
15.0			Prerequisites	Outcomes

See also

• UTSA Undergraduate Mathematics Course Descriptions