lect13_2

lect13_2 - Mathematical Induction. Induction is the most...

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Mathematical Induction. Induction is the most important proof method in computer science. Suppose you want to prove that the proposition P ( n ) is true for all n N , where N = {0, 1, 2, …} (an infinite set). ( i. e. every natural number n has some property P ) You might be able to prove it for 0, 1, 2, … But you can’t check one-by-one that all natural numbers have property P.
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The key idea of mathematical induction is start with 0 and repeatedly add 1. Suppose you can show that i) 0 has property P and ii) whenever you add 1 to a number that has property P the resulting number also has property P. This guarantees that as you go through the list of all natural numbers, every number you encounter must have property P .
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The proof method of mathematical induction says that you just need to do two things: i) prove it for n = 0 (basis case) ii) prove “if it’s true for n=k, then it’s true for n =k+ 1” Induction Hypothesis fix some k 0 assume P ( k ) Induction Step Using assumption P ( k ) prove that P ( k +1) In this way we prove that for any n [ P ( n ) P ( n +1)]
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lect13_2 - Mathematical Induction. Induction is the most...

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