(iii) Consider the statement, also from Chapter 16 exercises: For sets A and B we have A + B if and only if (A\B) U (B\A
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- Iii Consider The Statement Also From Chapter 16 Exercises For Sets A And B We Have A B If And Only If A B U B A 1 (118.95 KiB) Viewed 27 times
(iii) Consider the statement, also from Chapter 16 exercises: For sets A and B we have A + B if and only if (A\B) U (B\A) +0. = Proof. Suppose that A = B. Then (A\B) U (B\A) = (A\A) U (A\A) = 0 UD = 0. If A + B, then there exists E A but r¢ B or there exists x E B but x & A. In the former we have x € A\B hence z E (A\B) U (B\A). Therefore (A\B) U (B\A) # . A similar reasoning proves the result in the latter possibility. 126 CHAPTER 18. HOW TO READ A PROOF Analyze the proof (and don't forget to consider the extreme case of empty sets!). Summary Apply the reading techniques. Break the proof into pieces. Identify the methods used. Find where the assumptions are used. Apply the proof to an example. Draw a picture. O Check the text. O Look for mistakes. Apply the proof to a non-example. Memorise by understanding.