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18+Slides--Induction

# 18+Slides--Induction - CS103 HO#18 Slides-Induction The...

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CS103 HO#18 Slides--Induction 4/15/11 1 The Principle of Mathematical Induction 1 2 3 4 5 Suppose that: --We have a numbered collection of dominos --Each domino is standing --The first domino is knocked over --For any positive integer k, if domino k is knocked over, it knocks over domino k + 1 Do all the dominos fall? The Principle of Mathematical Induction 1 2 3 4 5 Suppose that: --We have a numbered collection of dominos --Each domino is standing --The first domino is knocked over --For any positive integer k, if domino k is knocked over, it knocks over domino k + 1 Let P n stand for the proposition: Domino n is knocked over. We are given: P 1 k(P k P k+1 ) And we conclude n P n The Principle of Mathematical Induction 1 2 3 4 5 Be sure you understand the notation: P 1 stands for the proposition: Domino 1 is knocked over. P 2 stands for the proposition: Domino 2 is knocked over. P 3 stands for the proposition: Domino 3 is knocked over. P 4 stands for the proposition: Domino 4 is knocked over. P 5 stands for the proposition: Domino 5 is knocked over. ... P k stands for the proposition: Domino k is knocked over. ... P n stands for the proposition: Domino n is knocked over. ... P n : You can reach rung n. What would convince you that you can reach all rungs of the ladder? P 1 : You can reach rung 1. k(P k P k+1 ) : If you can reach rung k, you can reach rung k + 1. If we can prove those two statements, then we can conclude: n P n P n : You can reach rung n. What would convince you that you can reach all rungs of the ladder? P 1 : You can reach rung 1. k(P k P k+1 ) : If you can reach rung k, you can reach rung k + 1. P n : You can reach rung n. What would convince you that you can reach all rungs of the ladder? P 7 : You can reach rung 7. k(P k P k+1 ) : If you can reach rung k, you can reach rung k + 1. If we can prove those two statements, then we can conclude: n 7 (P n )

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CS103 HO#18 Slides--Induction 4/15/11 2 P n : You can reach rung n. What would convince you that you can reach all rungs of the ladder? P 2 : You can reach rung 2. k(P k P k + 2 ) : If you can reach rung k, you can reach rung k + 2. P n : You can reach rung n. What would convince you that you can reach all rungs of the ladder? P 2 : You can reach rung 2. k(P k P k + 2 ) : If you can reach rung k, you can reach rung k + 2. Instead , we would renumber the rungs we can reach 1, 2, ... then use ordinary induction. Or , define R k = P 2k and do induction on the R's The Principle of Mathematical Induction A proof by mathematical induction that a proposition P n is true for every positive integer n consists of two steps: BASE CASE: Show that P 1 is true. INDUCTIVE STEP: Assume that P k is true for an arbitrarily chosen positive integer k and show that under that assumption , P k+1 must be true. From these two steps we conclude (by the principle of mathematical induction) that for all positive integers n, P n is true.
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18+Slides--Induction - CS103 HO#18 Slides-Induction The...

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