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Discrete Structure lecture_24

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DISCREAT STRUCTURE LECTURE # 24 PROOF BY CONTRADICTION KHAWAJA TARIQ MAHMOOD Lecture # 24 PROOF BY CONTRADICTION PROOF BY CONTRACTION: A proof by contradiction is based on the fact that either a statement is true or it is false but not both. Hence the supposition, that the statement to be proved is false, leads logically 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 a given 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: 2 Give a proof by contradiction for the statement: “If n is an even integer then n is an even integer.” SOLUTION: 2 Suppose n is an even integer and n is not even, so that n is odd. Hence n = 2k + 1 for some integer k. Now 2 n = (2k + 1) 2 2 = 4k + 4k + 1 2 = 2·(2k + 2k) + 1 = 2r + 1 2 2 where r = (2k + 2k) Z 2 This shows that n is odd, which is a contradiction to our supposition that n is even. Hence the given statement is true. EXAMPLE: Khawaja Tariq Mahmood. Page 2 Lecture # 24 PROOF BY CONTRADICTION 3 Prove 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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