Mathematical Induction. Part Two

Transcription

1 Mathematical Induction Part Two

2 Announcements Problem Set due Friday, January 8 at the start of class. Problem Set checkpoints graded, will be returned at end of lecture. Afterwards, will be available in the filing cabinets in the Gates Open Area.

3 The principle of mathematical induction states that if for some property P(n), we have that If it starts P(0) is true and For any n N, we have P(n) P(n + ) Then and it keeps going then it's always true. For any n N, P(n) is true.

4 n(n+ ) Theorem: For any natural number n, i= Proof: By induction. Let P(n) be 2 n n(n+ ) P(n) i= i= 2 For our base case, we need to show P(0) is true, meaning that 0 0(0+ ) i= i= 2 Since the empty sum is defined to be 0, this claim is true. n+ i= n i= n i= For the inductive step, assume that for some n N that P(n) holds, so n(n+ ) i= 2 We need to show that P(n + ) holds, meaning that n+ (n+ )(n+ 2) i= i= 2 To see this, note that n n(n+ ) n(n+ )+ 2(n+ ) (n+ )(n+ 2) i= i+ (n+ )= + n+ = = i= Thus P(n + ) is true, completing the induction.

5 Induction in Practice Typically, a proof by induction will not explicitly state P(n). Rather, the proof will describe P(n) implicitly and leave it to the reader to fill in the details. Provided that there is sufficient detail to determine what P(n) is, that P(0) is true, and that whenever P(n) is true, P(n + ) is true, the proof is usually valid.

6 Theorem: For any natural number n, Proof: By induction on n. For our base case, if n = 0, note that and the theorem is true for 0. For the inductive step, assume that for some n the theorem is true. Then we have that n(n+ ) n(n+ )+ 2(n+ ) (n+ )(n+ 2) i+ (n+ )= + n+ = = n+ i= n i= i= 0 i= i= so the theorem is true for n +, completing the induction. n i= 0(0+ ) =0 2 i= n(n+ ) 2

7 A Variant of Induction

8 n 2 versus 2 n 0 2 = 0 2 = 2 2 = = = = = = = = = 00 < < = > = < < < < < < 2 0 = 2 = = = = = = = = = = n n is is much much bigger bigger here. here. Does Does the the trend trend continue? continue?

9 Theorem: For any natural number n 5, n 2 < 2 n. Proof: By induction on n. As a base case, if n = 5, then we have that 5 2 = 25 < 32 = 2 5, so the claim holds. For the inductive step, assume that for some n 5, that n 2 < 2 n. Then we have that (n + ) 2 = n 2 + 2n + Since n 5, we have (n + ) 2 = n 2 + 2n + < n 2 + 2n + n (since < 5 n) = n 2 + 3n < n 2 + n 2 (since 3n < 5n n 2 ) = 2n 2 So (n + ) 2 < 2n 2. Now, by our inductive hypothesis, we know that n 2 < 2 n. This means that (n + ) 2 < 2n 2 (from above) < 2(2 n ) (by the inductive hypothesis) = 2 n + Completing the induction.

10 Induction Starting at k To prove that P(n) is true for all natural numbers greater than or equal to k: Show that P(k) is true. Show that for any n k, that P(n) P(n + ). Conclude P(k) holds for all natural numbers greater than or equal to k.

11 An Important Observation

12 One Major Catch In In an an inductive inductive proof, proof, to to prove prove P(5), P(5), we we can can only only assume assume P(4). P(4). We We cannot cannot rely rely on on any any of of our our earlier earlier results! results!

13 Strong Induction

14 The principle of strong induction states that if for some property P(n), we have that Assume Assume that that P(n) P(n) holds holds for for all all natural natural numbers numbers smaller smaller than than n. n. P(0) is true and For any n N with n 0, if P(0), P(),, and P(n ) are true, then P(n) is true then For any n N, P(n) is true.

15 Induction and Dominoes

16 Strong Induction and Dominoes

17 Weak and Strong Induction Weak induction (regular induction) is good for showing that some property holds by incrementally adding in one new piece. Strong induction is good for showing that some property holds by breaking a large structure down into multiple small pieces.

18 Why Use Weak Induction? Any proof done by weak induction can be done by strong induction. It is never wrong to use strong induction. However, if you only need the immediately previous result, weak induction can be a lot cleaner. More on that later on.

19 The principle of strong induction states that if for some property P(n), we have that Assume Assume that that P(n) P(n) holds holds for for all all natural natural numbers numbers smaller smaller than than n. n. P(0) is true and For any n N with n 0, if P(n') is true for all n' N satisfying 0 n' < n, then P(n) is true then For any n N, P(n) is true.

20 Proof by Strong Induction State that you are attempting to prove something by strong induction. State what your choice of P(n) is. Prove the base case: State what P(0) is, then prove it. Prove the inductive step: State that you assume for all 0 n' < n, that P(n') is true. State what P(n) is. (this is what you're trying to prove) Go prove P(n).

21 The Candy Bar Problem C A N D Y!

22 The Candy Bar Problem You are given a candy bar with n pieces. How many breaks do you have to make to separate it into n individual pieces?

23 n = 0 0 Breaks n = Break n = 2 2 Breaks n = 3 3 Breaks

24 n pieces k pieces k breaks n - k pieces n k breaks

25 Theorem: Breaking a chocolate bar with n pieces into individual pieces requires n breaks. Proof: By strong induction. Let P(n) be breaking a chocolate bar with n pieces into individual pieces requires n breaks. We prove P(n) holds for all n N with n. For our base case, we prove P(), that breaking a candy bar with one piece into individual pieces takes zero breaks. Since the candy bar is already in individual pieces, no breaks are required, so P() holds. For the inductive step, assume that for some n N with n, that for any n' N with n' < n, that P(n') is true and any candy bar with n' pieces requires n' breaks. We will prove P(n), that breaking any candy bar with n pieces requires n breaks. To see this, consider any possible break made in the candy bar. This will split the candy bar into two pieces, one of size k and one of size n k, where k < n. Note that n k < n as well, since the second piece is not empty and is not the entire candy bar. Since k < n, by our inductive hypothesis, breaking the k-piece candy bar requires k breaks. Since n k < n, by our inductive hypothesis, breaking the (n k)-piece candy bar takes n k breaks. Counting all breaks, we need k + n k + = n breaks for the n-piece candy bar. Thus P(n) holds, completing the induction.

26 Application: Continued Fractions

27 Continued Fractions

28 Continued Fractions

29 Continued Fractions A continued fraction is an expression of the form a + a 2 + a a n Formally, a continued fraction is either An integer n, or n + / F, where n is an integer and F is a continued fraction. Continued fractions have numerous applications in number theory and computer science. (They're also really fun to write!)

30 Fun with Continued Fractions Every rational number, including negative rational numbers, has a continued fraction representation. Harder result: every irrational number has an (infinite) continued fraction representation. Even harder result: If we truncate an infinite continued fraction for an irrational number, we can get progressively better approximations of that number.

31 π as a Continued Fraction π=

32 Approximating π π= = And And he he made made the the Sea Sea of of cast cast bronze, bronze, ten ten cubits cubits from from one one brim brim to to the the other; other; it it was was completely completely round. round. [ [ A] A] line line of of thirty thirty cubits cubits measured measured its its circumference. Kings Kings 7:23, 7:23, New New King King James James Translation Translation

33 Approximating π π= = /7 = Greek Greek mathematician Archimedes knew knew of of this this approximation, circa circa BCE BCE

34 Approximating π π= = /7 = /06 = /3 = Chinese Chinese mathematician 祖沖之祖沖之 (Zu (Zu Chongzhi) Chongzhi) discovered discovered this this approximation in in the the early early fifth fifth century; century; this this was was the the best best approximation of of π for for over over a a thousand thousand years. years.

35 Approximating π π= = /7 = /06 = /3 = /3302 =

36 More Continued Fractions =

37 An Interesting Continued Fraction x=

38 An Interesting Continued Fraction x= / 2 / 3 / 2 5 / 3 8 / 5 3 / 8 Each Each fraction fraction is is the the ratio ratio of of consecutive consecutive Fibonacci Fibonacci numbers! numbers! + 2 / 3 34 / 2

39 The Golden Ratio ϕ= = ϕ

40 The Golden Spiral

41 How do we prove all rational numbers have continued fractions?

42 Constructing a Continued Fraction = = = 3+ 2

43 Constructing a Continued Fraction = > 7 > 2 > 2

44 The Division Algorithm The division algorithm is the following theorem: For any integers a and b, with b > 0, there exists unique integers q and r such that and a = qb + r 0 r < b q is the quotient and r is the remainder. Given a = and b = 4: = Given a = -37 and b = 42: -37 =

45 The Division Algorithm n d = q+ d r n = qd + r

46 Theorem: Every rational has a continued fraction. Proof: By strong induction. Let P(d) be any rational with denominator d has a continued fraction. We prove that P(d) is true for all positive natural numbers. Since all rationals can be written with a positive denominator, this proves that all rationals have continued fractions. For our base case, we prove P(), that any rational with denominator has a continued fraction. Consider any rational with denominator ; let it be n /. Since n is a continued fraction and n = n /, P() holds. For our inductive step, assume that for some d N with d >, that for any d' N where d' < d, that P(d') is true, so any rational with denominator d' has a continued fraction. We prove P(d) by showing that any rational with denominator d has a continued fraction. Take any rational with denominator d; let it be n / d. Using the division algorithm, write n = qd + r, where 0 r < d. We consider two cases: Case : r = 0. Then n = qd, so n / d = q. Then q is a continued fraction for n / d. n Case 2: r 0. Given that n = qd + r, we have d =q+ r d =q+ d /r. Since r < d, by our inductive hypothesis there is some continued fraction for d / r; call it F. Then q + / F is a continued fraction for n / d. In either case, we find a continued fraction for n / d, so P(d) holds, completing the induction.

47 For more on continued fractions:

48 When Use Strong Induction? Normal (weak) induction is good for when you are shrinking the problem size by exactly one. Peeling one final term off a sum. Making one weighing on a scale. Considering one more action on a string. Strong induction is good when you are shrinking the problem, but you can't be sure by how much. Breaking a candy bar into two arbitrary smaller pieces. Taking the remainder of one number divided by another.

49 Next Time Graphs and Relations Representing structured data. Categorizing how objects are connected.