0. Communicating Mathematics

0.1 Learning Mathematics

0.2 What Others Have Said About Writing

0.3 Mathematical Writing

0.4 Using Symbols

0.5 Writing Mathematical Expressions

0.6 Common Words and Phrases in Mathematics

0.7 Some Closing Comments About Writing

**1. Sets**

1.1. Describing a Set

1.2. Subsets

1.3. Set Operations

1.4. Indexed Collections of Sets

1.5. Partitions of Sets

1.6. Cartesian Products of Sets

Chapter 1 Supplemental Exercises

**2. Logic**

2.1. Statements

2.2. The Negation of a Statement

2.3. The Disjunction and Conjunction of Statements

2.4. The Implication

2.5. More On Implications

2.6. The Biconditional

2.7. Tautologies and Contradictions

2.8. Logical Equivalence

2.9. Some Fundamental Properties of Logical Equivalence

2.10. Quantified Statements

2.11. Characterizations of Statements

Chapter 2 Supplemental Exercises

3. Direct Proof and Proof by Contrapositive

3.1. Trivial and Vacuous Proofs

3.2. Direct Proofs

3.3. Proof by Contrapositive

3.4. Proof by Cases

3.5. Proof Evaluations

Chapter 3 Supplemental Exercises

4. More on Direct Proof and Proof by Contrapositive

4.1. Proofs Involving Divisibility of Integers

4.2. Proofs Involving Congruence of Integers

4.3. Proofs Involving Real Numbers

4.4. Proofs Involving Sets

4.5. Fundamental Properties of Set Operations

4.6. Proofs Involving Cartesian Products of Sets

Chapter 4 Supplemental Exercises

5. Existence and Proof by Contradiction

5.1. Counterexamples

5.2. Proof by Contradiction

5.3. A Review of Three Proof Techniques

5.4. Existence Proofs

5.5. Disproving Existence Statements

Chapter 5 Supplemental Exercises

**6. Mathematical Induction**

6.1 The Principle of Mathematical Induction

6.2 A More General Principle of Mathematical Induction

6.3 Proof By Minimum Counterexample

6.4 The Strong Principle of Mathematical Induction

Chapter 6 Supplemental Exercises

**7. Reviewing Proof Techniques**

7.1 Reviewing Direct Proof and Proof by Contrapositive

7.2 Reviewing Proof by Contradiction and Existence Proofs

7.3 Reviewing Induction Proofs

7.4 Reviewing Evaluations of Proposed Proofs

Chapter 7 Supplemental Exercises

**8. Prove or Disprove**

8.1 Conjectures in Mathematics

8.2 Revisiting Quantified Statements

8.3 Testing Statements

Chapter 8 Supplemental Exercises

**9. Equivalence Relations**

9.1 Relations

9.2 Properties of Relations

9.3 Equivalence Relations

9.4 Properties of Equivalence Classes

9.5 Congruence Modulo n

9.6 The Integers Modulo n

Chapter 9 Supplemental Exercises

**10. Functions**

10.1 The Definition of Function

10.2 The Set of All Functions From A to B

10.3 One-to-one and Onto Functions

10.4 Bijective Functions

10.5 Composition of Functions

10.6 Inverse Functions

10.7 Permutations

Chapter 10 Supplemental Exercises

**11. Cardinalities of Sets**

11.1 Numerically Equivalent Sets

11.2 Denumerable Sets

11.3 Uncountable Sets

11.4 Comparing Cardinalities of Sets

11.5 The Schröder - Bernstein Theorem

Chapter 11 Supplemental Exercises

**12. Proofs in Number Theory**

12.1 Divisibility Properties of Integers

12.2 The Division Algorithm

12.3 Greatest Common Divisors

12.4 The Euclidean Algorithm

12.5 Relatively Prime Integers

12.6 The Fundamental Theorem of Arithmetic

12.7 Concepts Involving Sums of Divisors

Chapter 12 Supplemental Exercises

**13. Proofs in Combinatorics**

13.1 The Multiplication and Addition Principles

13.2 The Principle of Inclusion-Exclusion

13.3 The Pigeonhole Principle

13.4 Permutations and Combinations

13.5 The Pascal Triangle

13.6 The Binomial Theorem

13.7 Permutations and Combinations with Repetition

Chapter 13 Supplemental Exercises

**14. Proofs in Calculus**

14.1 Limits of Sequences

14.2 Infinite Series

14.3 Limits of Functions

14.4 Fundamental Properties of Limits of Functions

14.5 Continuity

14.6 Differentiability

Chapter 14 Supplemental Exercises

**15. Proofs in Group Theory**

15.1 Binary Operations

15.2 Groups

15.3 Permutation Groups

15.4 Fundamental Properties of Groups

15.5 Subgroups

15.6 Isomorphic Groups

Chapter 15 Supplemental Exercises

**16. Proofs in Ring Theory (Online)**

16.1 Rings

16.2 Elementary Properties of Rings

16.3 Subrings

16.4 Integral Domains

16.5 Fields

Chapter 16 Supplemental Exercises

**17. Proofs in Linear Algebra (Online)**

17.1 Properties of Vectors in 3-Space

17.2 Vector Spaces

17.3 Matrices

17.4 Some Properties of Vector Spaces

17.5 Subspaces

17.6 Spans of Vectors

17.7 Linear Dependence and Independence

17.8 Linear Transformations

17.9 Properties of Linear Transformations

Chapter 17 Supplemental Exercises

**18. Proofs with Real and Complex Numbers (Online)**

18.1 The Real Numbers as an Ordered Field

18.2 The Real Numbers and the Completeness Axiom

18.3 Open and Closed Sets of Real Numbers

18.4 Compact Sets of Real Numbers

18.5 Complex Numbers

18.6 De Moivre's Theorem and Euler's Formula

Chapter 18 Supplemental Exercises

**19. Proofs in Topology (Online)**

19.1 Metric Spaces

19.2 Open Sets in Metric Spaces

19.3 Continuity in Metric Spaces

19.4 Topological Spaces

19.5 Continuity in Topological Spaces

Chapter 19 Supplemental Exercises

Answers and Hints to Odd-Numbered Section Exercises

References

Index of Symbols

Index