This text makes a great supplement and provides a systematic approach for teaching undergraduate and graduate students how to read, understand, think about, and do proofs. The approach is to categorize, identify, and explain (at the student's level) the various techniques that are used repeatedly in all proofs, regardless of the subject in which the proofs arise. How to Read and Do Proofs also explains when each technique is likely to be used, based on certain key words that appear in the problem under consideration. Doing so enables students to choose a technique consciously, based on the form of the problem.
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
