How to Read and Do Proofs (Paperback, 6, Revised)

Foreword xi

Preface to the Student xiii

Preface to the Instructor xv

Acknowledgments xviii

Part I Proofs

1 Chapter 1: The Truth of It All 1

2 The Forward-Backward Method 9

3 On Definitions and Mathematical Terminology 25

4 Quantifiers I: The Construction Method 41

5 Quantifiers II: The Choose Method 53

6 Quantifiers III: Specialization 69

7 Quantifiers IV: Nested Quantifiers 81

8 Nots of Nots Lead to Knots 93

9 The Contradiction Method 101

10 The Contrapositive Method 115

11 The Uniqueness Methods 125

12 Induction 133

13 The Either/Or Methods 145

14 The Max/Min Methods 155

15 Summary 163

Part II Other Mathematical Thinking Processes

16 Generalization 179

17 Creating Mathematical Definitions 197

18 Axiomatic Systems 219

Appendix A Examples of Proofs from Discrete Mathematics 237

Appendix B Examples of Proofs from Linear Algebra 251

Appendix C Examples of Proofs from Modern Algebra 269

Appendix D Examples of Proofs from Real Analysis 287

Solutions to Selected Exercises 305

Glossary 357

References 367

Index 369