1. Induction
Chapter

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

3. 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.

4. 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

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

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

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

8. Example
Prove an inequity
Basis:
Induction:

9. 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:

10. 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

11. 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

12. Example: Postage Available stamps:
5¢ ¢
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)¢.

13. 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¢

14. 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¢