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Unformatted text preview: COT 3100 Spring 2010 Quiz #4 1. ( 4 pts ) What is wrong with the following proof? Theorem : For every positive integer , if ? and ? are positive integers with max ? , ? = , then ? = ? . Solution : Let P(n) be the proposition that if ? and ? are positive integers with max ? , ? = , then ? = ? . Basis Step : P(1) is true, since if max ? , ? = 1 then ? = ? = 1 . This completes the basis step. Inductive Step : Suppose that P(k) is true. That is if positive integers ? , ? satisfy max ? , ? = , then ? = ? . We will prove that P(k+1) is true. That is we must show that if positive integers ? , ? satisfy max ? , ? = + 1 , then ? = ? . Let max ? , ? = + 1 , then max ? 1, ? 1 = k, by the inductive hypothesis, ? 1 = ? 1 . It follows that ? = ? , completing the inductive step. 2. ( 6 pts ) Prove that for any natural number > 2 , 1 1 2 1 1 3 1 1 < 2 2 Solution: 1. The inductive step is wrong. The proof fails for P(2), that is the proposition if The inductive step is wrong....
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This note was uploaded on 01/22/2012 for the course COT 3100 taught by Professor Staff during the Spring '08 term at University of Florida.
- Spring '08