MAT3013

From Department of Mathematics at UTSA
Jump to navigation Jump to search

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

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
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
1.0
  • 1.1
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
1.0
  • 1.1
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
1.0
  • 1.1
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
2.0
  • 1.2-3
  • Gain the prerequisites for writing and evaluating proofs.
2.0
  • 1.2-3
  • Gain the prerequisites for writing and evaluating proofs.
2.0
  • 1.2-3
  • Gain the prerequisites for writing and evaluating proofs.
2.0
  • 1.2-3
  • Gain the prerequisites for writing and evaluating proofs.
3.0
  • 1.4-6
  • Start proving elementary results.
4.0
  • 2.1-3
  • How to start working with sets
4.0
  • 2.1-3
  • How to start working with sets
4.0
  • 2.1-3
  • How to start working with sets
5.0
  • 2.4-6
  • Learn constructive proofs and reasoning.
  • Learn basic counting principles of discrete mathematics.
5.0
  • 2.4-6
  • 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
  • Gain basic concepts about relations.
7.0
  • 3.1-3
  • Gain basic concepts about relations.
7.0
  • 3.1-3
  • Gain basic concepts about relations.
8.0
  • 3.4-5
  • Familiarize with ordering.
  • Learn how to use graph representations of relations.
8.0
  • 3.4-5
  • Familiarize with ordering.
  • Learn how to use graph representations of relations.
9.0
  • 4.1-2
  • Gain basic rigorous knowledge of functions.
9.0
  • 4.1-2
  • Gain basic rigorous knowledge of functions.
10.0
  • 4.3-4
  • Determine whether a function is one-to-one with proofs.
10.0
  • 4.3-4
  • Determine whether a function onto with proofs.
10.0
  • 4.3-4
  • Finding inverses
11.0
  • 4.5-6
  • Find images of subsets under functions, with proofs.
11.0
  • 4.5-6
  • Find preimages of subsets under functions, with proofs.
11.0
  • 4.5-6
12.0
  • Catch up and review
  • Midterm 2
13.0
  • 5.1-2
  • Learn classification of sets by size.
13.0
  • 5.1-2
  • Learn classification of sets by size.
  • Generalizing the concept of size to infinite sets
13.0
  • 5.1-2
14.0
  • 5.3-5
  • Learn properties of countable sets.
14.0
  • 5.3-5
  • 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
1.0
  • 1.1-1.2
  • Historical remarks
  • Overview of the course and its goals
  • Ideas of proofs and logic
  • Logical statements
  • Motivation for rigorous mathematics from a historical perspective
  • An understanding of where and why this course is going
2.0
  • 1.1-1.5
  • Statements
  • Relation between statements
  • Valid Arguments
  • Quantifiers
  • Prerequisites
  • Outcomes
3.0
  • 2.2-2.4
  • Why we need proofs
  • Direct proofs
  • Proofs by contrapositive and contradiction
  • Cases and If and Only If
  • Prerequisites
  • Outcomes
4.0
  • 2.6, 3.1-3.3
  • Basic concepts
  • Operations and constructions with sets
  • Prerequisites
  • Outcomes
5.0
  • 3.4-3.5
  • Family of sets
  • Axioms of set theory
  • Prerequisites
  • Outcomes
6.0
7.0
  • 4.1-4.3
  • Definition of functions
  • Image and inverse image
  • Composition and inverse functions
  • Prerequisites
  • Outcomes
8.0
  • 4.4-4.5
  • Injectivity, surjectivity and bijectivity
  • Sets of functions
  • Prerequisites
  • Outcomes
9.0
  • 5.1-5.2
  • Relations
  • Congruence
  • Prerequisites
  • Outcomes
10.0
  • 4.3-4
  • Equivalence relations
  • Prerequisites
  • Outcomes
11.0
  • 6.1-6.2
  • Introduction
  • Properties of natural numbers
  • Prerequisites
  • Outcomes
12.0
13.0
  • 6.2-6.3
  • Mathematical induction
  • Recursion
  • Prerequisites
  • Outcomes
14.0
  • 6.4 - 6.7
  • Cardinality of sets
  • Finite sets and countable sets
  • Cardinality of number systems
  • Prerequisites
  • Outcomes
15.0

See also