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How to Read and Do Proofs

Daniel Solow
John Wiley
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
William Porter
, on

Daniel Solow’s How to Read and Do Proofs begins with the simpler methods of mathematical proof-writing and gradually works toward the more advanced techniques typically presented in an introduction to advanced mathematics.

This book accomplishes the vast majority of what it was written to do. Solow develops, with careful detail and numerous examples, all of the proof techniques typically introduced in a proof-writing class. Starting with the basic idea of logically manipulating both sides of the proposition in an attempt to connect them, he works up to more subtle and more powerful techniques involving uniqueness, methods of induction, and proof by contrapositive . Every chapter offers examples of proofs using that chapter’s namesake technique, each of which is followed by a step-by-step analysis.

A strong point in this book’s favor is that it offers so many examples from so many different fields of mathematics. The last hundred pages or so are devoted to examples from particular subjects, including linear and abstract algebra, analysis, and set theory. Throughout the book, the author presents many ubiquitous mathematical definitions. An introduction to such a wide range of mathematics early in the curriculum will provide up-and-coming math majors with a more accurate idea of their future studies, helping to dispel the freshman fear that all mathematics is like Calculus II. The exercises are quite varied and offer a great test of the student’s understanding.

The explanation of proof by contradiction and the description of induction are wonderful. They are extremely intuitive and provide the reader with a satisfying understanding of how these tools function.

Not everything is so intuitive though. Some of the proofs feel a bit contrived, the logic a bit obscure. This gap may be indicative of the maturity that comes with mathematical experience. Proofs by construction don’t always seem obvious.

The language that Solow creates to describe general proof methods at times becomes hard to read. Some of the sentences seem as though they should have been spoken rather than read; maybe some italics are needed to set off longer phrases e.g. “the something that happens” is often used to stand for a generic outcome following the phrase “suppose there is an object such that.” When this phrase comes up every other sentence, it creates wordy, long-winded dialogue.

Definitions are sometimes used in exercises before the chapters in which they are introduced. This mix-up is just confusing for an experienced reader, but could really be a problem for a beginner, particularly when the purpose of the book is to teach precise logical argumentation.

This book is a very solid, detailed introduction to mathematical proof-writing. It would function well as the main text if the focus of the class was writing proofs, and it would serve excellently as a supplement to classes that offer proof-writing as a side goal: a course in discrete mathematics, an introductory class in set theory, or a basic course in topology.

William Porter is an undergraduate mathematics-physics double major at a small liberal arts college. He enjoys ballroom dancing, cooking, and T. S. Eliot’s poetry. He thinks that Rachmaninoff writes amazing music and rain is the best weather. His favorite Big Bang Theory character is Sheldon.

1. The Truth of It All.

2. The Forward-Backward Method

3. On Definitions and Mathematical Terminology.

4. Quantifiers I: The Construction Method.

5. Quantifiers II: The Choose Method.

6. Quantifiers III: Specialization.

7. Quantifiers IV: Nested Quantifiers.

8. Nots of Nots Lead to Knots.

9. The Contradiction Method.

10. The Contrapositive Method.

11. The Uniqueness Methods

12. Induction.

13. The Either/Or Methods.

14. The Max/Min Methods.

15. Summary.

Appendix A: Examples of Proofs from Discrete Mathematics.

Appendix B: Examples of Proofs from Linear Algebra.

Appendix C: Examples of Proofs from Modern Algebra.

Appendix D: Examples of Proofs from Real Analysis.

Solutions to Selected Exercises.