# Discrete Structure lecture_23

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 # 23METHOD OF PROOFKHAWAJA TARIQ MAHMOODLecture # 23METHOD OF PROOFMETHOD OF PROOF:To understand written mathematics, one must understand what makes up a correctmathematical argument, that is, a proof. This requires an understanding of the techniquesused to build proofs. The methods we will study for building proofs are also usedthroughout computer science, such as the rules computers used to reason, the techniquesused to verify that programs are correct, etc.METHODS OF PROOFINDIRECT PROOFDIRECT PROOFpqPROOF BYCONTRAPOSITIONp q ~q~pPROOF BYCONTRADICTIONp q (p~q) cDIRECT PROOF:The implication p q can be proved by showing that if p is true, the q must also betrue. This shows that the combination p true and q false never occurs. A proof of this kindis called a direct proof.pqpqTTFFTFTFTFTTKhawaja Tariq Mahmood.Page 2Lecture # 23METHOD OF PROOFEXAMPLE:Prove that the sum of two odd integers is evenSOLUTION:Let m and n be two odd integers. Then by definition of odd numbersm = 2k + 1for some k Zn = 2l + 1for some l ZNow m + n = (2k + 1) + (2l + 1)= 2k + 2l + 2= 2 (k + l + 1)= 2rwhere r = (k + l + 1) ZHence m + n is even.EXAMPLE:Prove that if n is any even integer, then(-1)n = 1SOLUTION:Suppose n is an even integer. Then n = 2k for some integer k.Now(-1) n= (-1)2k= [(-1)2]k= (1)k=1(proved)EXAMPLE:Prove that the product

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