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How to Read and Do Proofs, provides a systematic approach for teaching students how to read, think about, understand, and create proofs. It develops a method for communicating proofs, categorizing, identifying, and explaining (at the student’s level) the various techniques that are used repeatedly in virtually all proofs. These clear, concise explanations promote understanding of the theoretical mathematics behind abstract mathematics and give students a greater opportunity to succeed in advanced courses. Along with the addition of three new chapters, a “Part 2” is added to the Sixth Edition, which focuses on the mathematical thought processes associated with proofs. The teaching of this foregoing thinking processes reduces the time needed for readers to learn advanced mathematics courses while simultaneously increasing their depth of understanding so as to enable them to use mathematics more effectively as a problem-solving tool in their personal and professional lives.
Part I Proofs
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