21 Induction. Tom Lewis. Fall Term Tom Lewis () 21 Induction Fall Term / 14

Transcription

1 21 Induction Tom Lewis Fall Term 2010 Tom Lewis () 21 Induction Fall Term / 14

2 Outline 1 The method of induction 2 Strong mathematical induction Tom Lewis () 21 Induction Fall Term / 14

3 Pessimists and Optimists In the solving of problems, the method of smallest counterexample is a glass-half-empty approach; mathematical induction is a glass-half-full approach. Tom Lewis () 21 Induction Fall Term / 14

4 Tom Lewis () 21 Induction Fall Term / 14

5 Tipping propositions with mathematical induction Tom Lewis () 21 Induction Fall Term / 14

6 Tipping propositions with mathematical induction Suppose that we want to show that each of the propositions P 0, P 1, P 2,... is true. Tom Lewis () 21 Induction Fall Term / 14

7 Tipping propositions with mathematical induction Suppose that we want to show that each of the propositions P 0, P 1, P 2,... is true. First show that P 0 is true. This is the customary base case or basis step. Tom Lewis () 21 Induction Fall Term / 14

8 Tipping propositions with mathematical induction Suppose that we want to show that each of the propositions P 0, P 1, P 2,... is true. First show that P 0 is true. This is the customary base case or basis step. Here is the tricky part. If we can show that for each k 0, P k is true }{{} The Induction Hypothesis = P k+1 is true. then, in fact, we will shown that each P k, k N, is true. Tom Lewis () 21 Induction Fall Term / 14

9 Problem Use mathematical induction to prove the following theorem. Tom Lewis () 21 Induction Fall Term / 14

10 Problem Use mathematical induction to prove the following theorem. Theorem Let n be a natural number. Then n = n(n + 1). 2 Tom Lewis () 21 Induction Fall Term / 14

11 Problem Use mathematical induction to prove the following theorem. Tom Lewis () 21 Induction Fall Term / 14

12 Problem Use mathematical induction to prove the following theorem. Theorem Let n be a natural number. Then n = 2 n+1 1 Tom Lewis () 21 Induction Fall Term / 14

13 Problem Use mathematical induction to prove the following theorem. Tom Lewis () 21 Induction Fall Term / 14

14 Problem Use mathematical induction to prove the following theorem. Theorem Let n 1 be an integer. Then n < 2 n. Tom Lewis () 21 Induction Fall Term / 14

15 Problem Use mathematical induction to prove the following theorem. Tom Lewis () 21 Induction Fall Term / 14

16 Problem Use mathematical induction to prove the following theorem. Theorem Let n be a natural number. Then 4 n 1 is divisible by 3. Tom Lewis () 21 Induction Fall Term / 14

17 Problem Let {a n } be a sequence of numbers defined recursively as: a 0 = 10 and, for n 1, a n = 1 2 a n Use mathematical induction to prove the following theorem. Tom Lewis () 21 Induction Fall Term / 14

18 Problem Let {a n } be a sequence of numbers defined recursively as: a 0 = 10 and, for n 1, a n = 1 2 a n Use mathematical induction to prove the following theorem. Theorem Let n be a natural number. Then a n 20. Tom Lewis () 21 Induction Fall Term / 14

19 Tiling with L-triominos Tom Lewis () 21 Induction Fall Term / 14

20 Theorem Let n 1 be an integer. A set of L-triominos can tile all but one cell of a 2 n 2 n chessboard. Tom Lewis () 21 Induction Fall Term / 14

21 Strong mathematical induction Tipping propositions with mathematical induction Tom Lewis () 21 Induction Fall Term / 14

22 Strong mathematical induction Tipping propositions with mathematical induction Suppose that we want to show that each of the propositions P 0, P 1, P 2,... is true. Tom Lewis () 21 Induction Fall Term / 14

23 Strong mathematical induction Tipping propositions with mathematical induction Suppose that we want to show that each of the propositions P 0, P 1, P 2,... is true. First show that P 0 is true. This is the customary base case or basis step. Tom Lewis () 21 Induction Fall Term / 14

24 Strong mathematical induction Tipping propositions with mathematical induction Suppose that we want to show that each of the propositions P 0, P 1, P 2,... is true. First show that P 0 is true. This is the customary base case or basis step. Here is the tricky part. If we can show that for each k 0, P 0, P 1,..., P k are true }{{} The Strong Induction Hypothesis = P k+1 is true. then, in fact, we will shown that each P k, k N, is true. Tom Lewis () 21 Induction Fall Term / 14

25 Strong mathematical induction Problem Define a sequence {g n } as follows: g 1 = 5, g 2 = 13, and g n = 5g n 1 6g n 2 n 3. Show that g n = 2 n + 3 n for all n 1. Tom Lewis () 21 Induction Fall Term / 14