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L142 Agenda Mathematical Induction Proofs Well Ordering Principle Simple Induction Strong Induction (Second Principle of Induction) Program Correctness Correctness of iterative Fibonacci program

L143 Mathematical Induction Suppose we have a sequence of propositions which we would like to prove: P (0), P (1), P (2), P (3), P (4), … P (n), … EG: P (n) = “The sum of the first n positive odd numbers is the n th perfect square” We can picture each proposition as a domino: P (n)

L144 Mathematical Induction So sequence of propositions is a sequence of dominos. … P (n+1)P (n) P (2)P (1)P (0)

L145 Mathematical Induction When the domino falls, the corresponding proposition is considered true: P (n)

L146 Mathematical Induction When the domino falls (to right), the corresponding proposition is considered true: P (n) true

L147 Mathematical Induction Suppose that the dominos satisfy two constraints. 1) Well-positioned: If any domino falls (to right), next domino (to right) must fall also. 2) First domino has fallen to right P (0) true P (n+1)P (n)

L148 Mathematical Induction Suppose that the dominos satisfy two constraints. 1) Well-positioned: If any domino falls to right, the next domino to right must fall also. 2) First domino has fallen to right P (0) true P (n+1)P (n)

L149 Mathematical Induction Suppose that the dominos satisfy two constraints. 1) Well-positioned: If any domino falls to right, the next domino to right must fall also. 2) First domino has fallen to right P (0) true P (n) true P (n+1) true

L1410 Mathematical Induction Then can conclude that all the dominos fall! … P (n+1)P (n) P (2)P (1)P (0)

L1411 Mathematical Induction Then can conclude that all the dominos fall! … P (n+1)P (n) P (2)P (1)P (0)

L1412 Mathematical Induction Then can conclude that all the dominos fall! …P (0) true P (n+1)P (n) P (2)P (1)

L1413 Mathematical Induction Then can conclude that all the dominos fall! …P (0) true P (1) true P (n+1)P (n) P (2)

L1414 Mathematical Induction Then can conclude that all the dominos fall! P (2) true …P (0) true P (1) true P (n+1)P (n)

L1415 Mathematical Induction Then can conclude that all the dominos fall! P (2) true …P (0) true P (1) true P (n+1)P (n)

L1416 Mathematical Induction Then can conclude that all the dominos fall! P (2) true …P (0) true P (1) true P (n) true P (n+1)

L1417 Mathematical Induction Then can conclude that all the dominos fall! P (2) true …P (0) true P (1) true P (n) true P (n+1) true

L1418 Mathematical Induction Principle of Mathematical Induction: If: 1) [basis] P (0) is true 2) [induction]  n P(n)  P(n+1) is true Then:  n P(n) is true This formalizes what occurred to dominos. P (2) true …P (0) true P (1) true P (n) true P (n+1) true

L1419 Mathematical Induction Example EG: Prove  n  0 P(n) where P(n) = “The sum of the first n positive odd numbers is the n th perfect square.” =

L1420 Mathematical Induction Example Every induction proof has two parts, the basis and the induction step. 1) Basis: Show that the statement holds for n = 1. In our case, plugging in 0, we would like to show that: 

L1421 Mathematical Induction Example 2) Induction: Show that if statement holds for k, then statement holds for k+1. (induction hypothesis)  This completes proof. 

More Examples In class notes L1422