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How To Read and Do Proofs

Daniel Solow
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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The principal change in this new edition of Solow's well-known text is the addition of four appendices with "Examples of Proofs" from Discrete Mathematics, Linear Algebra, Modern Algebra, and Real Analysis. This probably reflects the feeling of many instructors that it makes little sense to teach students to do proofs out of the context of some specific part of mathematics. The appendices presumably allow instructors to choose one such topic and choose examples accordingly.

Solow's book attempts to teach students how to construct and understand proofs by creating a taxonomy of proof techniques and teaching them to students both by description and by example. Parts of this taxonomy are standard (e.g., "proof by contrapositive"), and parts seem to be Solow's own creation (e.g., "forward uniqueness").

The author stresses, and reviewers in the past have agreed, that this book will only be useful in the context of a course or seminar that emphasizes class participation. In particular, they claim that assigning it as supplementary reading would not be productive. I tend to agree, but also note that teaching from this book requires the instructor to "buy into" Solow's taxonomy and approach in a big way.

Solow's textbook is probably one of the best for instructors who want to approach the "Transition to Proofs" course emphasizing proof techniques rather than putting the focus on logic or undertaking a survey of advanced mathematics.

 Fernando Q. Gouvêa teaches at Colby College in Waterville, ME.


Preface to the Student.

Preface to the Instructor.


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. Uniqueness Methods and Induction.

12. Either/or and Max/Min Methods.

13. 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 Select Exercises.