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How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, 5th Edition
John Wiley and Sons Ltd, Jan 2010, Pages: 301
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.
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 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 happens.”
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 methods.
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.
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