How to Read and Do Proofs

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

"The instructional material is to the point, with well-considered examples and asides on common mistakes. Good examples of the author's thoughtfulness appear in the discourses on pp. 5-6 of identifying the hypothesis and conclusion when they are not obvious, on pp. 28-29 regarding overlapping notation, and on pp. 190-191 of the advantages and disadvantages of generalization." (Zentralblatt MATH 2016)

Daniel Solow is a professor of management for the Weatherhead School of Management at Case Western Reserve University. His research interests include developing and analyzing optimization models for studying complex adaptive systems, and basic research in deterministic optimization, including combinatorial optimization, linear and nonlinear programming. He has published over 20 papers on both topics.