MAT3013

Foundations of Mathematics (3-0) 3 Credit Hours

Course Catalog

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

Prerequisite: MAT 1214. 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 MAT 2243. Credit cannot be earned for MAT 3013 and MAT 2243.) 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 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
2.0	1.2-3	Logical operators Truth values Truth tables Quantifiers		Gain the prerequisites for writing and evaluating proofs.
3.0	1.4-6	Methods for proofs	Propositional logic	Start proving elementary results.
4.0	2.1-3	Basic concepts Operations and constructions with sets	Basic concepts of set theory	How to start working with sets
5.0	2.4-6	Mathematical induction Counting principles	Natural numbers	Learn constructive proofs and reasoning. Learn basic counting principles of discrete mathematics.
6.0
7.0	3.1-3	Cartesian products and their subsets Equivalence relations	Set theory	Gain basic concepts about relations.
8.0	3.4-5	Partial orders Graphs	Relations 1	Familiarize with ordering. Learn how to use graph representations of relations.
9.0	4.1-2	Functions Constructions with functions	Relations Function sense (precalculus)	Gain basic rigorous knowledge of functions.
10.0	4.3-4	One-to-one Onto Compositional inverse	Functions 1	Determine whether a function is one-to-one of onto, with proofs. Finding inverses
11.0	4.5-6	Images of subsets Preimages of subsets Sequences	Functions 2	Find images and preimages of subsets under functions, with proofs.
12.0
13.0	5.1-2	Finite and infinite sets Equivalent sets	Sets and functions	Learn classification of sets by size. Generalizing the concept of size to infinite sets
14.0	5.3-5	Countable and uncountable sets	Cardinality 1	Learn properties of countable sets.
15.0

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

• UTSA Undergraduate Mathematics Course Descriptions