How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, 5th Edition
December 2009, ©2009
This text provides a systematic approach for teaching undergraduate and graduate students how to read, understand, think about, and do proofs. The approach is to catagorize, 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. Students are taught how to read proofs that arise in textbooks and other mathematical literature by understanding which techniques are used and how they are applied. It shows how any proof can be understood as a sequence of the individual techniques. The goal is to enable students to learn advanced mathematics on their own. This book is suitable as: (1) a text for a transition-to-advanced-math course, (2) a supplement to any course involving proofs, and (3) self-guided teaching.
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
13. The Either/Or Methods.
14. The Max/Min Methods.
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.
What’s New in the Fifth Edition
The main change in the fifth edition is a complete revision and expansion of
the exercises in the main body of the text. This book now contains exercises
that are appropriate for all levels of undergraduate students. As in the fourth
edition, all exercises marked with a B have completely worked-out solutions
in the back of the book; those marked with a W have complete solutions on
the web at http://www.wiley.com/college/solow/; and the rest have solutions
provided in the accompanying Solutions Manual that only instructors can
obtain from the foregoing web site. Exercises marked with a * symbol and
whose solution is not available to students are considered relatively more
challenging or time-consuming.
Other changes in this edition include the following:
1. A discussion in Chapter 1 of the need to identify the hypothesis and
conclusion when the proposition is not stated in the standard form,
“If A, then B.” Several examples are given to illustrate how this is done
and appropriate exercises are included.
2. An extended and more complete discussion in Chapter 3 of how to use
a previously-proved proposition in both the forward and backward processes.
3. A discussion in Chapter 5 of the equivalence of the statements, “For all
objects X with a certain property, something happens” and “If X is
an object with a certain property, then X satisfies the something that
4. Replacing the previous Chapters 11 and 12 with new Chapters 11 - 14
so as to devote a separate self-contained chapter with exercises to each of
the following techniques: uniqueness, induction, either/or, and max/min
5. The inclusion of several final examples of how to read and do proofs in
the summary Chapter 15 that serve to unify the student’s knowledge of
the various proof techniques.
Although these changes seem to make it even easier for students to understand
proofs, I have still found no substitute for actively teaching the material
in class instead of having the students read the material on their own. This
active interaction has proved eminently beneficial to both student and teacher,
in my case.