20a-Mathematical-Induction

20a-Mathematical-Induction - Mathematical Introduction...

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Mathematical Induction Discrete Mathematics
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Discrete Mathematics – Mathematical Induction 20-2 Principle of Mathematical Induction Climbing an infinite ladder We can reach the first rung For all k, standing on the rung k we can step on the rung k + 1 Can we reach every step of it, if
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Discrete Mathematics – Mathematical Induction 20-3 Principle of Mathematical Induction Principle of mathematical induction : To prove that a statement that assert that some property P(n) is true for all positive integers n, we complete two steps Basis step : We verify that P(1) is true. Inductive step : We show that the conditional statement P(k) P(k + 1) is true for all positive integers k Symbolically, the statement (P(1) k (P(k) P(k + 1))) n P(n) How do we do this? P(1) is usually an easy property To prove the conditional statement, we assume that P(k) is true (it is called inductive hypothesis ) and show that under this assumption P(k + 1) is also true
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Discrete Mathematics – Mathematical Induction 20-4 The Domino Effect Show that all dominos fall: Basis Step: The first domino falls Inductive step: Whenever a domino falls, its next neighbor will also fall
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Discrete Mathematics – Mathematical Induction 20-5 Summation Prove that the sum of the first n natural numbers equals that is P(n): `the sum of the first n natural numbers … Basis step: P(1) means Inductive step: Make the inductive hypothesis, P(k) is true, i.e. Prove P(k + 1):
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Discrete Mathematics – Mathematical Induction 20-6 More Summation Prove that Let P(n) be the statement for the integer n Basis step: P(0) is true, as Inductive step: We assume the inductive hypothesis and prove P(k + 1), that is We have
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Discrete Mathematics – Mathematical Induction 20-7 No Horse of Different Color Prove that in any herd of n horses all horses all horses are of the same color Basis step: n =1, clear
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20a-Mathematical-Induction - Mathematical Introduction...

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