Principle of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))

Transcription

1 Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction

2 Principle of (Weak) Mathematical Induction P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))

3 Principle of (Weak) Mathematical Induction P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n)) Alternate view: P(1) ( n 2)(P(n 1) P(n)) ( n 1)(P(n))

4 Requirements Mathematical Inductive proofs must have:

5 Requirements Mathematical Inductive proofs must have: Base case: P(1) Usually easy

6 Requirements Mathematical Inductive proofs must have: Base case: P(1) Usually easy Inductive hypothesis: Assume P(n 1)

7 Requirements Mathematical Inductive proofs must have: Base case: P(1) Usually easy Inductive hypothesis: Assume P(n 1) Inductive step: Prove P(n 1) P(n)

8 Arithmetic series: A first example Example For all n 1 n 4i 2 = (4n 6) + (4n 2) i=1 = 2n 2 Do in class.

9 Arithmetic series: Gauss s sum Example For all n 1 n i = (n 1) + n i=1 = n(n + 1) 2 Do in class.

10 Sum of powers of 2 Example For all n 1 n 1 2 k = n n 1 k=0 = 2 n 1 Do in class.

11 Geometric Series Example For all n 1 and real r 1 n 1 r k = 1 + r + r 2 + r r n 2 + r n 1 k=0 = rn 1 r 1 Do in class.

12 A divisibility property Example For all integers n 0 n 3 n (mod 3) Do in class. Use P(n) P(n + 1) and/or start with Induction Hypothesis.

13 Size of Power Set Theorem Let A be a finite set. Then P(A) = 2 A Proof in class.

14 A recurrence relation Example Let Then a n = { a n 1 + (2n 1) if n 2 1 if n = 1 a n = n 2 for n 1 Do in class.

15 An inequality Example For all integers n 3 2n + 1 < 2 n Do in class.

16 Another inequality Example For all integers n 0 and real x nx (1 + x) n Do in class.

17 Catalan Numbers Example C n = ( ) 1 2n n + 1 n = (2n)! n!(n + 1)! For all integers n 1 (2n)! n!(n + 1)! 4 n (n + 1) 2 Also on handout. Do in class.

18 A less mathematical example Example If all we have is 2 cent and 5 cent coins, we can make change for any amount of money at least 4 cents. Do in class.

19 A recurrence relation Example Start with a 0 = 1. a 1 = a = = 2. a 2 = (a 0 + a 1 ) + 1 = (1 + 2) + 1 = 4. a 3 = (a 0 + a 1 + a 2 ) + 1 = ( ) + 1 = 8. a 4 = (a 0 +a 1 +a 2 +a 3 )+1 = ( )+1 = 16. In general, a n = = 2 n ( n 1 ) a i + 1 = (a 0 + a 1 + a a n 1 ) + 1 i=0

20 A recurrence relation Example Start with a 0 = 1. a 1 = a = = 2. a 2 = (a 0 + a 1 ) + 1 = (1 + 2) + 1 = 4. a 3 = (a 0 + a 1 + a 2 ) + 1 = ( ) + 1 = 8. a 4 = (a 0 +a 1 +a 2 +a 3 )+1 = ( )+1 = 16. In general, a n = ( n 1 ) a i + 1 = (a 0 + a 1 + a a n 1 ) + 1 i=0 = 2 n Proof on next slide!!!

21 Proof of recurrence relation by mathematical induction Theorem { 1 if n = 0 a n = ( n 1 ) i=0 a i + 1 = a 0 + a a n if n 1 Then a n = 2 n. Proof by Mathematical Induction. Base case easy. Induction Hypothesis: Assume a n 1 = 2 n 1. Induction Step: a n = ( n 1 ) a i + 1 = i=0 ( n 2 ) a i + a n Now what? i=0

22 Proof of recurrence relation by mathematical induction Theorem { 1 if n = 0 a n = ( n 1 ) i=0 a i + 1 = a 0 + a a n if n 1 Then a n = 2 n. Proof by Mathematical Induction. Base case easy. Induction Hypothesis: Assume a n 1 = 2 n 1. Induction Step: a n = ( n 1 ) a i + 1 = i=0 ( n 2 ) a i + a n Now what? i=0 = (a n 1 1) + a n = 2a n 1 = 2 2 n 1 = 2 n.

23 Principle of Strong (Mathematical) Induction Recall weak mathematical induction: P(1) ( n 2)(P(n 1) P(n)) ( n 1)(P(n))

24 Principle of Strong (Mathematical) Induction Recall weak mathematical induction: P(1) ( n 2)(P(n 1) P(n)) ( n 1)(P(n)) Strong mathematical induction: P(1) ( n 2)(P(1) P(2) P(n 1) P(n)) ( n 1)(P(n))

25 Proof of recurrence relation by strong induction Theorem { 1 if n = 0 a n = ( n 1 ) i=0 a i + 1 = a 0 + a a n if n 1 Then a n = 2 n. Proof by Strong Induction. Base case easy. Induction Hypothesis: Assume a i = 2 i for 0 i < n. Induction Step: a n = ( n 1 ) a i + 1 = i=0 ( n 1 = (2 n 1) + 1 = 2 n. ) 2 i + 1 i=0

26 Another recurrence relation Example Let 1 if n = 0 1 if n = 1 a n = 3 if n = 2 a n 1 + a n 2 a n 3 if n 3 Then a n is odd, for n 0. Do in class.

27 Yet another recurrence relation Example Let Then for all integers i 0 0 if i = 0 a i = 7 if i = 1 2a i 1 + 3a i 2 if i 2 a i 0 (mod 7) Do in class.

28 And yet another recurrence relation Example Let 0 if i = 1 a i = 2 if i = 2 3a i 2 if i 3 Then for all integers i 1, a i is even. Do in class.

29 Size of prime numbers Example Let p n be the nth prime number. Then p n 2 2n Proof in class.

30 Jigsaw Puzzle How many moves does it take to put together a jigsaw puzzle with n pieces? Do in class.

31 Principle of Constructive Induction If you know or guess the form of the answer, you can sometimes derive the actual answer while doing mathematical induction to prove it.

32 Constructive Induction: Example Example For all n 1 n 4i 2 =? i=1 Guess that for all integers n 1, n 4i 2 = an 2 + bn + c i=1 Why? Find constants a, b, and c such that this holds. Do in class.

33 Constructive induction: Recurrence Example Let 2 if n = 0 a n = 7 if n = 1 12a n 1 + 3a n 2 if n 2 What is a n? Guess that for all integers n 0, a n AB n Why? Find constants A and B such that this holds: Primarily find smallest B and secondarily smallest A. Do in class.

34 Structural Induction Definition (Loosely speaking...) Structural induction is strong induction over recursively defined objects.

35 A geometric example Definition A triangulated polygon is a decomposition of the polygon into triangles by non-intersecting lines. Example in class.

36 A geometric example Definition A triangulated polygon is a decomposition of the polygon into triangles by non-intersecting lines. Example in class. Not recursively defined.

37 A geometric example Definition A triangulated polygon is a decomposition of the polygon into triangles by non-intersecting lines. Example in class. Definition (Alternative recursive version) A triangulated polygon is Either a triangle, Not recursively defined. or a polygon with a straight line drawn between two vertices (that are not next to each other), where the two polygons formed by this line and the original polygon are themselves triangulated polygons.

38 A geometric example continued Definition A coloring of a triangulated polygon is an assignment of colors to all of the vertices of the polygon so that no two vertices that share an edge have the same color. Example in class. Theorem Every triangulated polygon is 3-colorable. Proof in class.

39 Full binary tree Definition A full binary tree is a rooted tree where every node has exactly zero or two children. Definition (Alternative recursive version) A full binary tree is Either a single node, called the root, or a single node, called the root, with exactly two children, where each child is the root of a full binary tree. Example in class.

40 Tree definitions Definition The distance between two nodes is the number of edges between them. Definition A leaf is a node with no children. An internal node is a node with children. Definition The external path length is the sum of the distances from the root to all of the leaves. The internal path length is the sum of the distances from the root to all of the internal nodes.

41 Internal and external path lengths Theorem Let N be the number of nodes in a full binary tree. Let E and I be its external and internal path lengths, respectively. Then E = I + N 1 Two proofs in class.