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	Prerequisites	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.
1/2	1.3-1.4	Semantic Implication Semantic 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.
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