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Introduction to Logic and Proof

Revised: September 2020

Course Description

An introduction to the principles of logic and the methods of proof necessary for the successful study of mathematics. Prerequisites MATH 140 or MATH 153. Three credit hours.

Student Learning Objectives

By the end of the course, students should be able to:

• Use the principles of logic, mathematical reasoning, and methods of proof writing to develop proofs for never-before-seen theorems and lemmas;
• Identify the most appropriate proof-writing technique for a given theorem or lemma based on mathematical setting;
• Demonstrate development in their mathematical writing through practice in communicating mathematical ideas and arguments, both verbally and orally.

Text

An Introduction to Abstract Mathematics, Robert J. Bond and William J. Keane, Waveland Press 1999.

Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

• Chapter 1 Mathematical Reasoning

• Section 1.1 Statements
• Section 1.2 Compound Statements

• Section 1.3 Implications

• Section 1.4 Contrapositive and Converse

• Chapter 2 Sets

• Section 2.1 Sets and Subsets
• Section 2.2 Combining Sets
• Section 2.3 Collections of Sets

• Chapter 3 Functions

• Section 3.1 Definitions and Basic Properties
• Section 3.2 Surjective and Injective Functions
• Section 3.3 Composition and Invertible Functions

• Chapter 4 Binary Operations and Relations

• Section 4.1 Binary Operations
• Section 4.2 Equivalence Relations

• Chapter 5 The Integers

• Section 5.1 Axioms and Basic Properties (optional)
• Section 5.2 Induction
• Section 5.3 The Division Algorithm and Greatest Common Divisors
• Section 5.4 Prime and Unique Factorization
• Section 5.5 Congruences

Additional topics selected at the instructor’s discretion.