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Mathematical Proofs: A Transition to Advanced Mathematics (Subscription), 3rd edition

Published by (October 12th 2012) - Copyright ©

3rd edition

Mathematical Proofs: A Transition to Advanced Mathematics (Subscription)

ISBN-13: 9780321892577

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Overview

Table of contents

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

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