Induction Chapter 4.1-4.2

Odd Powers Are Odd If m is odd and n is odd, then nm is odd Proposition: for an odd number m, mk is odd for all non-negative integer k. Let P(i) be the proposition that mi is odd. Proof by induction

Proof by induction P(1) is true by definition. P(2) is true by P(1) and the fact. P(3) is true by P(2) and the fact. P(i+1) is true by P(i) and the fact. So P(i) is true for all i.

Mathematical Induction 1 and (from n to n +1) are valid, then, proves 1, 2, 3,…. Expressed as a rule of inference To prove that P(n) is true for all positive integers, where P(n) is a propositional function Two steps: Basis: verify P(1) is true Show that conditional statement if P(k) then P(k+1) is true for all positive integers k

Induction Basis step: Very easy to prove Induction step: Much easier to prove with P(n) as an assumption Domino effects.

Proof by Induction Example: summation of integers Proof: P(1) is true Assume: Then

Example Prove: Basis: Induction: if Then add to both side rn+1

Example Prove an inequity Basis: Induction:

Strong Induction To prove that P(n) is true for all positive integers n, where P(n) is a propositional function Tow steps: Basis step: verify that P(1) is true Inductive step: show that the conditional statement is true for ALL positive integers k Conclusion:

Equivalent to Ordinary Induction From ordinary induction: 0 to 1, 1 to 2, 2 to 3, …, n-1 to n So by the time we got to n+1, already know all of P(0), P(1), …, P(n) So strong induction is equivalent to ordinary one

Example: Prime Products Claim: Every integer > 1 is a product of primes. Proof:(by strong induction) Base case is easy, 2 is a prime itself; Suppose the claim is true for all 2 <= i < n. Consider an integer n; In particular, n is not prime since if it is we do not need to prove. So n = k·m for integers k, m where n > k,m>1. Since k,m smaller than n, by the induction hypothesis, both k and m are product of prime Therefore, n is a prime product

Example: Postage Available stamps: 5¢ 3¢ What amount can we get from these stamps Theorem: Can form any amount larger or equal to 8¢ Prove by strong induction on all n larger or equal to 8 P(n) ::= can form (n+8)¢.

Postage: Strong induction proof Basis step: n=0 (0+8)¢=8¢. Inductive Step: assume (m +8)¢ for 0 <= m < n then prove ((n +1) + 8)¢ Case 1: n +1=1, 9¢ Case 2: n +1=2, 10¢

Postage: Strong induction proof Case 3: n +1 >= 3: let m=n-2 now n >= m >= 0, so by induction hypothesis + = (n+1)+8 can be formed (n-2)+8 is formed Add a 3¢