Discrete Structure lecture_24

Apr 28th, 2015
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It includes all the details about the Discrete structure with examples.Step by step solution.

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DISCREAT STRUCTURELECTURE # 24PROOF BYCONTRADICTIONKHAWAJA TARIQ MAHMOODLecture # 24PROOF BY CONTRADICTIONPROOF BY CONTRACTION:A proof by contradiction is based on the fact that either a statement is true or itis false but not both. Hence the supposition, that the statement to be proved is false, leadslogically to a contradiction, impossibility or absurdity, then the supposition must be false.Accordingly, the given statement must be true.This method of proof is also known as reductio ad absurdum because it relies on reducing agiven assumption to an absurdity.The method of proof by contradiction, may be summarized as follows:1.Suppose the statement to be proved is false.2.Show that this supposition leads logically to a contradiction.3.Conclude that the statement to be proved is true.EXAMPLE:2Give a proof by contradiction for the statement: If n is an even integer then n isan even integer.SOLUTION:2Suppose n is an even integer and n is not even, so that n is odd.Hence n = 2k + 1 for some integer k.Now2n = (2k + 1)22= 4k + 4k + 12= 2(2k + 2k) + 1= 2r + 122where r = (2k + 2k) Z2This shows that n is odd, which is a contradiction to our supposition that n is even. Hence thegiven statement is true.EXAMPLE:Khawaja Tariq Mahmood.Page 2Lecture # 24PROOF BY CONTRADICTION3Prove that if n is an integer and n + 5 is odd, then n is even using contradiction method.SOLUTION:Suppose that n3 + 5 is odd

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