strongInductionExamples

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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Strong Mathematical Induction Strong Mathematical Induction
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 2
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)
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example: Divisibility by a Prime Example: Divisibility by a Prime Proof ( cont .): We have that for all i Z with 2≤i<k,
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 6

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online