Mathematical Proofs Book

0. Communicating Mathematics

Learning Mathematics

What Others Have Said About Writing

Mathematical Writing

Using Symbols

Writing Mathematical Expressions

Common Words and Phrases in Mathematics

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

Exercises for Chapter 1

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

Exercises for Chapter 2

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

Exercises for Chapter 3

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

Exercises for Chapter 4

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

Exercises for Chapter 5

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

Exercises for Chapter 6

7. Prove or Disprove

7.1 Conjectures in Mathematics

7.2 Revisiting Quantified Statements

7.3 Testing Statements

Exercises for Chapter 7

8. Equivalence Relations

8.1 Relations

8.2 Properties of Relations

8.3 Equivalence Relations

8.4 Properties of Equivalence Classes

8.5 Congruence Modulo n

8.6 The Integers Modulo n

Exercises for Chapter 8

9. Functions

9.1 The Definition of Function

9.2 The Set of All Functions from A to B

9.3 One-to-one and Onto Functions

9.4 Bijective Functions

9.5 Composition of Functions

9.6 Inverse Functions

9.7 Permutations

Exercises for Chapter 9

10. Cardinalities of Sets

10.1 Numerically Equivalent Sets

10.2 Denumerable Sets

10.3 Uncountable Sets

10.4 Comparing Cardinalities of Sets

10.5 The Schröder-Bernstein Theorem

Exercises for Chapter 10

11. Proofs in Number Theory

11.1 Divisibility Properties of Integers

11.2 The Division Algorithm

11.3 Greatest Common Divisors

11.4 The Euclidean Algorithm

11.5 Relatively Prime Integers

11.6 The Fundamental Theorem of Arithmetic

11.7 Concepts Involving Sums of Divisors

Exercises for Chapter 11

12. Proofs in Calculus

12.1 Limits of Sequences

12.2 Infinite Series

12.3 Limits of Functions

12.4 Fundamental Properties of Limits of Functions

12.5 Continuity

12.6 Differentiability

Exercises for Chapter 12

13. Proofs in Group Theory

13.1 Binary Operations

13.2 Groups

13.3 Permutation Groups

13.4 Fundamental Properties of Groups

13.5 Subgroups

13.6 Isomorphic Groups

Exercises for Chapter 13

14. Proofs in Ring Theory (Online)

14.1 Rings

14.2 Elementary Properties of Rings

14.3 Subrings

14.4 Integral Domains

14.5 Fields

Exercises for Chapter 14

15. Proofs in Linear Algebra (Online)

15.1 Properties of Vectors in 3-Space

15.2 Vector Spaces

15.3 Matrices

15.4 Some Properties of Vector Spaces

15.5 Subspaces

15.6 Spans of Vectors

15.7 Linear Dependence and Independence

15.8 Linear Transformations

15.9 Properties of Linear Transformations

Exercises for Chapter 15

16. Proofs in Topology (Online)

16.1 Metric Spaces

16.2 Open Sets in Metric Spaces

16.3 Continuity in Metric Spaces

16.4 Topological Spaces

16.5 Continuity in Topological Spaces

Exercises for Chapter 16

Answers and Hints to Odd-Numbered Section Exercises

References

Index of Symbols

Index of Mathematical Terms