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

• 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

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
1.	1.1-1.2	Statements Sentential Logic		Identify syntactically correct formulas in sentential logic. Translate compound statements in informal language to formal propositional sentences. Find the interpretation of a sentential formula given interpretations of the propositional symbols therein.
2.	1.3-1.4	Logical Implication Logical Equivalence Deductive Rules	Sentential Logic	Express informally stated relations between sentences in terms of semantic implication and equivalence. State and recognize basic rules of deductive reasoning and their correct application. Use the rules of deduction to prove basic semantic relations (implication or equivalence) between formal interpretations of propositional formulas. Distinguish between correct and incorrect applications of deductive rules.
3.	1.5-2.2	Quantifiers Mathematical Proofs Proofs:Direct	Sentential Logic Deductive Rules
4.	2.3-2.4	Proofs:Contraposition Proofs:Contradiction Proofs:Cases	Mathematical Proofs Proofs:Direct
5.	2.4-2.6	Proofs:Biconditionals Proofs:Quantifiers Writing Mathematics	Quantifiers Mathematical Proofs
6.	3.1-3.3	Sets:Definitions Sets:Operations Sets:Families
7.		Review of Chapters 1-3. Midterm exam.
8.	4.1-4.3	Functions:Definition Functions:Forward Image Functions:Inverse Image Functions:Composition	Sets:Definitions Sets:Operations
9.	4.3-4.4	Functions:Inverses Functions:Injective Functions:Surjective Functions:Bijective	Functions:Definition Functions:Composition
10.	5.1 & 5.3	Relations Functions as Relations Equivalence Relations	Sets:Definitions Sets:Operations
11.	6.1-6.3	Natural Numbers:Postulates Natural Numbers:Well-Ordering Proofs:Induction	Sets:Definitions Functions:Definition Relations
12.	6.3-6.4	Induction:Variants Recursion	Proofs:Induction Functions:Definition
13.	6.5	Sets:Cardinality	Sets:Definitions Equivalence Relations Functions:Injective Functions:Bijective
14.	6.6-6.7	Sets:Finite Sets:Countable Sets:Uncountable Cardinality:𝐍 Cardinality:𝐙 Cardinality:𝐐 Cardinality:𝐑	Sets:Cardinality Natural Numbers:Postulates
15.		Catch-up and review for final exam.

See also

• UTSA Undergraduate Mathematics Course Descriptions