strongInductionExamples

# strongInductionExamples - P(i) i is divisible by a prime...

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1 Strong Mathematical Induction Strong Mathematical Induction

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Principle of  Principle of  Strong Mathematical Induction Strong Mathematical Induction Let P(n) be a predicate defined for integers n; a and b be fixed integers with a≤b . Suppose the following statements are true: 1. P(a), P(a+1), … , P(b) are all true (basis step) 2. For any integer k>b, if P(i) is true for all integers i with a≤i<k, then P(k) is true. (inductive step) Then P(n) is true for all integers n≥a.
3 Example: Divisibility by a Prime Example: Divisibility by a Prime Theorem: For any integer n≥2, n is divisible by a prime. P(n) Proof ( by strong mathematical induction ): 1) Basis step: The statement is true for n=2 P(2) because 2 | 2 and 2 is a prime number . 2) Inductive step: Assume the statement is true for all i with 2≤i<k P(i) (inductive hypothesis) ; show that it is true for k . P(k)

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Example: Divisibility by a Prime Example: Divisibility by a Prime Proof ( cont .): We have that for all i Z with 2≤i<k,

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Unformatted text preview: P(i) i is divisible by a prime number. (1) We must show: P(k) k is also divisible by a prime. (2) Consider 2 cases: a) k is prime . Then k is divisible by itself. b) k is composite . Then k=ab where 2a<k and 2b<k. Based on (1), p|a for some prime p. p|a and a|k imply that p|k (by transitivity). Thus, P(n) is true by strong induction. Proving a Property of a Sequence Proving a Property of a Sequence Proposition: Suppose a , a , a , is defined as follows: Proving a Property of a Sequence Proving a Property of a Sequence Proof ( cont. ): 2) Inductive step: For any k>2 , Assume P(i) is true for all i with 0i<k: a i 2 i for all 0i<k . (1) Show that P(k) is true: a k 2 k (2) a k = a k-1 +a k-2 +a k-3 2 k-1 +2 k-2 +2 k-3 (based on (1)) 2 +2 1 ++2 k-3 +2 k-2 +2 k-1 = 2 k-1 (as a sum of geometric sequence) 2 k Thus, P(n) is true by strong induction....
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## This note was uploaded on 05/25/2008 for the course MATH 21-127 taught by Professor Gheorghiciuc during the Fall '07 term at Carnegie Mellon.

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strongInductionExamples - P(i) i is divisible by a prime...

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