Cover

LIST OF SYMBOLS

Core Topics and Section Prerequisites

Title Page

Copyright

Contents

Preface

Preface to the Student

Chapter 1 Logic and Proofs

1.1 Propositions and Connectives

1.2 Conditionals and Biconditionals

1.3 Quantified Statements

1.4 Basic Proof Methods Ⅰ

1.5 Basic Proof Methods Ⅱ

1.6 Proofs Involving Quantifiers

1.7 Strategies for Constructing Proofs

1.8 Proofs from Number Theory

Chapter 2 Sets and Induction

2.1 Basic Concepts of Set Theory

2.2 Set Operations

2.3 Indexed Families of Sets

2.4 Mathematical Induction

2.5 Equivalent Forms of Induction

2.6 Principles of Counting

Chapter 3 Relations and Partitions

3.1 Relations

3.2 Equivalence Relations

3.3 Partitions

3.4 Modular Arithmetic

3.5 Ordering Relations

Chapter 4 Functions

4.1 Functions as Relations

4.2 Constructions of Functions

4.3 Functions That Are Onto;One-to-One Functions

4.4 Inverse Functions

4.5 Set Images

4.6 Sequences

4.7 Limits and Continuity of Real Functions

Chapter 5 Cardinality

5.1 Equivalent Sets;Finite Sets

5.2 Infinite Sets

5.3 Countable Sets

5.4 The Ordering of Cardinal Numbers

5.5 Comparability and the Axiom of Choice

Chapter 6 Concepts of Algebra

6.1 Algebraic Structures

6.2 Groups

6.3 Subgroups

6.4 Operation Preserving Maps

6.5 Rings and Fields

Chapter 7 Concepts of Analysis

7.1 The Completeness Property

7.2 The Heine–Borel Theorem

7.3 The Bolzano–Weierstrass Theorem

7.4 The Bounded Monotone Sequence Theorem

7.5 Equivalents of Completeness

Appendix Sets,Number Systems,and Functions

Answers to Selected Exercises

Index