Strong Mathematical Induction

1. Strong Mathematical Induction Lecture 23 Section 4.4 Tue, Feb 27, 2007

2. Announcements • Thursday at 4:00 (cookies at 3:30).

3. The Principle of Strong Mathematical Induction • Let P(n) be a predicate defined for integers n. Let a be an integer. If the following two statements are true • P(a), P(a + 1), …, P(b). • For all integers kb, if P(a), P(a + 1), …, P(k), then P(k + 1). then the statement • For all integers na, P(n) is true.

4. The Principle of Strong Mathematical Induction • The range a, a + 1, a + 2, …, b represents the number of previous cases that the inductive step depends on. • Usually this is 1. • We will see one example where it is 2.

5. Example: Factoring Integers • Theorem: Every integer n 2 factors into primes. • Proof: • Base case: • Let n = 2. Then n is already a prime, so the statement is true.

6. Example: Factoring Integers • Inductive case: • Suppose the statement is true for all integers from 2 to k, for some k  2. • Consider k + 1. • Either k + 1 is prime or it is not prime. • If it is prime, then we are done. • If it is not prime, then it factors as a  b, for some integers a, b, with a, b  2.

7. Example: Factoring Integers • By the induction hypothesis, a and b themselves factor into the product of primes. • Therefore, k + 1 factors into the product of primes.

8. Example: The Fibonacci Sequence • The Fibonacci sequence {fn} is defined by • f0 = 0, • f1 = 1, • fn = fn – 1 + fn – 2, for all n 2. • The first few terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

9. Example: The Fibonacci Sequence • Theorem: Let Then for all n  0.

10. Example: The Fibonacci Sequence • Proof: • Base cases: • Let n = 0. Then (0 – 0)/( – ) = 0. • Let n = 1. Then (1 – 1)/( – ) = 1. • Inductive case: • Suppose the equation holds for all n from 0 to k, for some k  1. • Consider fk + 1.

11. A Lemma • Lemma: n + 1 = n + n – 1 and n + 1 = n + n – 1 for all n  1. • Proof (for ): • Base case: • Let n = 1. • Then a direct calculation shows that 2 =  + 1 = 1 + 0.

12. A Lemma • Inductive step: • Now suppose it is true for some integer k  1. • That is, k + 1 = k + k – 1 • Multiply the equation through by . • The result is k + 2 = k + 1 + k. • So the statement is true when n = k + 1. • Therefore, the statement is true for all n  1.

13. Example: The Fibonacci Sequence • Back to our theorem…

14. Example: The Fibonacci Sequence • Therefore, the statement is true for all n 1.

15. Example: Trees • A tree is a connectedgraph that contains no cycles.

16. Example: Trees • Notice that if you add an edge to a tree, you necessarily create a cycle. • Also notice that if you delete an edge from a tree, the graph becomes disconnected. • This suggests that a tree contains exactly the “right” number of edges.

17. Example: Trees • Theorem: For all n  1, a tree with nnodes contains exactly n – 1 edges. • Proof: • Base case: • Let n = 1. • Clearly, there are 0 edges.

18. Example: Trees • Inductive case: • Suppose that the statement is true for all n k, for some k 1. • Let T be a tree with k + 1 nodes. • Let v be an arbitrary node in T and let s be the index of v. • Let u1, u2, …, us be the adjacent nodes. • Delete the edges from v to u1, …, us.

19. Example: Trees • Then each ui is a node of a separate tree Ti. • Let ni be the number of nodes in tree Ti. • Note that n1 + … + ns = k. • By induction, each tree Ti contains ni – 1 edges. • Thus, the total number of edges in tree T is (n1 – 1) + … + (ns – 1) + s, which equals k.

20. Example: Trees

21. Example: Trees u2 v u3 u1

22. Example: Trees u2 v u3 u1

23. Example: Trees u2 v u3 u1 4 nodes, 3 edges

24. Example: Trees 1 node, 0 edges u2 v u3 u1 4 nodes, 3 edges

25. Example: Trees 1 node, 0 edges u2 v u3 u1 4 nodes, 3 edges 3 nodes, 2 edges

26. Example: Trees 1 node, 0 edges u2 v u3 u1 4 nodes, 3 edges 3 nodes, 2 edges 3 edges

27. Example: Trees u2 v u3 u1 Total = 3 + 0 + 2 + 3 = 8 edges

28. Example: Trees

29. Example: Trees u2 u1 v u3

30. Example: Trees u2 u1 v u3

31. Example: Trees u2 u1 v 6 nodes, 5 edges u3

32. Example: Trees 1 node, 0 edges u2 u1 v 6 nodes, 5 edges u3

33. Example: Trees 1 node, 0 edges u2 u1 v 6 nodes, 5 edges u3 1 nodes, 0 edges

34. Example: Trees 1 node, 0 edges u2 u1 v 6 nodes, 5 edges u3 1 nodes, 0 edges 3 edges

35. Example: Trees u2 u1 v u3 Total = 5 + 0 + 0 + 3 = 8 edges

36. Example: Binary Trees • A binary tree structure consists of a set of nodes, each with two pointers. • The left pointer points to the left child. • The right pointer points to the right child. • If there is no child, then the pointer is null.

37. Example: Binary Trees • Corollary: In a binary tree structure, if there are n nodes, then there are exactly n + 1 null pointers.

38. Balanced Binary Strings • Let S be the set of all finite binary strings that contain an equal number of 0’s and 1’s. • Define a set T as follows: • The empty string is in T. • If s  T, then 0s1  T and 1s0  T. • If s  T and t  T, then st  T.

39. Balanced Binary Strings • Theorem: T S. • Theorem: T = S.

40. The Paradox of the Pop Quiz • A professor announces that one day during the semester he will give a pop quiz. • For the quiz to be a “pop” quiz, it must be unexpected.

41. The Paradox of the Pop Quiz • Theorem: For all n 0, the professor cannot give the pop quiz n days before the last day of the semester.

42. The Paradox of the Pop Quiz • Proof: • Base case: n = 0. • He cannot give the pop quiz on the last day of the semester because everyone will be expecting it then.

43. The Paradox of the Pop Quiz • Inductive step: • Suppose he cannot give it on any of the days 0, 1, 2, …, k days before the end of the semester, for some k 0. • Knowing this, the students would be expecting it on the day k + 1 days before the end of the semester. • So he can’t give it on that day.

44. The Paradox of the Pop Quiz • By strong induction, for all n 0, he cannot give the pop quiz n days before the end of the semester.