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47 Slides--Induction III, Fibonacci

# 47 Slides--Induction III, Fibonacci - CS103A HO 47...

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CS103A HO# 47 Induction III, Fibonacci 2/27/08 1 CI WOP Well Ordering Property: If A is a non-empty set of positive integers, then A has a least element. CI WOP Well Ordering Property: If A is a non-empty set of positive integers, then A has a least element. Proof. Suppose that A is a set of positive integers without a least element, and let P(n) be the proposition that n A. We will show that n P(n) by complete mathematical induction, i.e., that A must be empty. BASE CASE: Since 1 is the smallest positive integer, 1 A, because if so, 1 would be the least element of A. So P(1) is true. INDUCTIVE STEP: Assume: for some positive integer k, P(i) for 1 i k. Show P(k + 1). CI WOP Well Ordering Property: If A is a non-empty set of positive integers, then A has a least element. Proof. Suppose that A is a set of positive integers without a least element, and let P(n) be the proposition that n A. We will show that n P(n) by complete mathematical induction, i.e., that A must be empty. BASE CASE: Since 1 is the smallest positive integer, 1 A, because if so, 1 would be the least element of A. So P(1) is true. INDUCTIVE STEP: Assume: for some positive integer k, P(i) for 1 i k. Show P(k + 1). If k + 1 A, k + 1 would be the least element in A, since no integer less than k + 1 is in A by the inductive hypothesis. Thus k + 1 A, and P(k+1) is true. Thus by C.I., n P(n). Since A is a set of positive integers to which no integer belongs, we have shown that if A has no least element, it must be empty. CI WOP Well Ordering Property: If A is a non-empty set of positive integers, then A has a least element. Proof. We will show that if A is a set of positive integers without a least element, then A is empty. Let P(n) be the proposition that n A. We will show that n P(n) by complete mathematical induction. BASE CASE: Since 1 is the smallest positive integer, 1 A, because if so, 1 would be the least element of A. So P(1) is true. INDUCTIVE STEP: Assume: for some positive integer k, P(i) for 1 i k.

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47 Slides--Induction III, Fibonacci - CS103A HO 47...

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