PART I

Chapter 1: LANGUAGE, LOGIC, AND SETS

1.1 Logic and Language

1.2 Implication

1.3 Quantifiers and Definitions

1.4 Introduction to Sets

1.5 Introduction to Number Theory

1.6 Additional Set Theory

Definitions from Chapter 1

Algebraic and Order Properties of Number Systems

Chapter 2: PROOFS

2.1 Proof Format I: Direct Proofs

2.2 Proof Format II: Contrapositive and Contradition

2.3 Proof Format III: Existence, Uniqueness, Or

2.4 Proof Format IV: Mathematical Induction

The Fundamental Theorem of Arithmetic

2.5 Further Advice and Practice in Proving

Proof Formats

Chapter 3: FUNCTIONS

3.1 Definitions

3.2 Composition, One-to-One, Onto, and Inverses

3.3 Images and Pre-Images of Sets

Definitions from Chapter 3

Chapter 4: RELATIONS

4.1 Relations

4.2 Equivalence Relations

4.3 Partitions and Equivalence Relations

4.4 Partial Orders

Definitions from Chapter 4

PART II

Chapter 5: INFINTE SETS

5.1 The Sizes of Sets

5.2 Countable Sets

5.3 Uncountable Sets

5.4 The Axiom of Choice and Its Equivalents

Definitions from Chapter 5

Chapter 6: INTRODUCTION TO DISCRETE MATHEMATICS

6.1 Graph Theory

6.2 Trees and Algorithms

6.3 Counting Principles I

6.4 Counting Principles II

Definitions from Chapter 6

Chapter 7: INTRODUCTION TO ABSTRACT ALGEBRA

7.1 Operations and Properties

7.2 Groups

Groups in Geometry

7.3 Rings and Fields

7.4 Lattices

7.5 Homomorphisms

Definitions from Chapter 7

Chapter 8: INTRODUCTION TO ANALYSIS

8.1 Real Numbers, Approximations, and Exact Values

Zeno’s Paradoxes

8.2 Limits of Functions

8.3 Continuous Functions and Counterexamples

Counterexamples in Rational Analysis

8.4 Sequences and Series

8.5 Discrete Dynamical Systems

The Intermediate Value Theorem

Definitions for Chapter 8

Chapter 9: METAMATHEMATICS AND THE PHILOSOPHY OF MATHEMATICS

9.1 Metamathematics

9.2 The Philosophy of Mathematics

Definitions for Chapter 9

Appendix: THE GREEK ALPHABET

Answers: SELECTED ANSWERS

Index

List of Symbols