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quiz_4 - COT 3100 Spring 2010 Quiz#4 1(4 pts What is wrong...

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COT 3100 Spring 2010 Quiz #4 1. ( 4 pts ) What is wrong with the following proof? Theorem : For every positive integer 𝑛 , if ? and ? are positive integers with max ? , ? = 𝑛 , then ? = ? . Solution : Let P(n) be the proposition that if ? and ? are positive integers with max ? , ? = 𝑛 , 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 ? , ? = 𝑘 , then ? = ? . We will prove that P(k+1) is true. That is we must show that if positive integers ? , ? satisfy max ? , ? = 𝑘 + 1 , then ? = ? . Let max ? , ? = 𝑘 + 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 𝑛 > 2 , 1 1 2 1 1 3 1 1 𝑛 < 2 𝑛 2
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Solution: 1. The inductive step is wrong. The proof fails for P(2), that is the proposition if ? , ? are positive and max ? , ? = 2 , then ? = ? is false. Given max( ? , ? ) = 2 , it is true that max ? − 1, ? − 1 = 1
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