The Foundations of Mathematics

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