Discrete Structure Lecture 22

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 # 22MATHEMATICALINDUCTIONKHAWAJA TARIQ MAHMOODLecture # 22MATHIMATICAL INDUCTIONPRINCIPLE OF MATHEMATICAL INDUCTION:Let P n be a propositional function defined for all positive integers n.P n is truefor every positive integer n if1. BASIC STEP:The proposition P 1 is true.2. INDUCTIVE STEP:If P k is true then P k 1 is true for all integers k 1.i.e. kP k P k 1EXAMPLE:Use Mathematical Induction to prove that1 2 3 .......... n n n 12for all integers n 1SOLUTION:LetP n 1 2 3 .......... n n n 121. Basis Step:P 1 is true.For n = 1, left hand side of P 1 is the sum of all the successive integers starting at 1 and endingat 1, so LHS = 1 and RHS isR.H .S 11 1 2 122So the proposition is true for n = 1.Khawaja Tariq Mahmood.Page 2Lecture # 22MATHIMATICAL INDUCTION2. Inductive Step:Suppose P k is true for, some integers k 1.k k 1..........(1)21 2 3 .......... k To prove P k 1 is true. That is,1 2 3 .......... k 1 1 2 3 .......... k 1 k 1 k 1 12 k 1 k 2 ..........(2)2Adding k 1 in equation (1) on both side, we get1 2 3 ..........

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