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New to This Edition
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The main change in the fourth edition is the replacement of the two previous appendices with four new ones in discrete mathematics, linear algebra, modern algebra, and real analysis. The purpose of these appendices is to illustrate how the various proof techniques from the body of the text arise in doing actual mathematics.
o The chapter on writing the negation of a statement includes a new section on counterexamples and now appears immedi-ately before the chapter on the contradiction and contrapositive methods.
o The backward uniqueness method in Chapter 11, for showing that there is a unique object with a certain property such that something happens, has been rewritten slightly to clarify the two-step process of showing that there is at least one ob-ject with the certain property and for which the something happens, and that there is at most one such object (using the direct and indirect uniqueness methods.
o The inclusion in Chapter 11 of a new forward uniqueness method that students can use when they encounter a unique object with a certain property such that something happens in the forward process, that is, when they know that there is a unique object with a certain property such that something happens.
o In addition to Chapter 13, a summary of all of the proof techniques appears inside the front and back covers for quick reference.
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