lecture induction2 6up

CS311H: Discrete Mathematics

Mathematical Induction

Isl Dillig

Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 1/26

Announcements

I Homework 5 out today

I Due next Thursday (Oct 30)


Review: Strong Induction

I Base case same as regular induction, different in inductive step

I Regular induction: assume P(k) holds and prove P(k + 1)

I Strong induction: assume P(1),P(2), ..,P(k); prove P(k + 1)


Example

I Prove that every integer n 12 can be written asn = 4a + 5b for some non-negative integers a, b.

I Proof by strong induction on n and consider 4 base cases

I Base case 1 (n=12): 12 = 3 4 + 0 5

I Base case 2 (n=13): 13 = 2 4 + 1 5

I Base case 3 (n=14): 14 = 1 4 + 2 5

I Base case 4 (n=15): 15 = 0 4 + 3 5


Example, cont.

Prove that every integer n 12 can be written as n = 4a + 5b forsome non-negative integers a, b.

I Inductive hypothesis: Suppose every 12 i k can bewritten as i = 4a + 5b.

I Inductive step: We want to show k + 1 can also be writtenthis way for k + 1 16

I Observe: k + 1 = (k 3) + 4

I By the inductive hypothesis, k 3 = 4a + 5b for some a, bbecause k 3 12

I But then, k + 1 can be written as 4(a + 1) + 5b


Matchstick Example

I The Matchstick game: There are two piles with same numberof matches initially

I Two players take turns removing any positive number ofmatches from one of the two piles

I Player who removes the last match wins the game

I Prove: Second player always has a winning strategy.


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Matchstick Proof

I P(n): Player 2 has winning strategy if initially n matches ineach pile

I Base case:

I Induction: Assume j .1 j k P(j ); show P(k + 1)

I Inductive hypothesis:

I Prove Player 2 wins if each pile contains k + 1 matches


Matchstick Proof, cont.

I Case 1: Player 1 takes k + 1 matches from one of the piles.

I What is winning strategy for player 2

I Case 2: Player 1 takes r matches from one pile, where1 r k

I Now, player 2 takes r matches from other pile

I Now, the inductive hypothesis applies player 2 has winningstrategy for rest of the game


The Horse Paradox

I Easy to make subtle errors when trying to prove things byinduction pay attention to details!

I Consider the statement: All horses have the same color

I What is wrong with the following bogus proof of thisstatement?

I P(n) : A collection of n horses have the same color

I Base case: P(1) X


Bogus Proof, cont.

I Induction: Assume P(k); prove P(k + 1)

I Consider a collection of k + 1 horses: h1, h2, . . . , hk+1

I By IH, h1, h2, . . . , hk have the same color; let this color be c

I By IH, h2, . . . , hk+1 have same color; call this color c

I Since h2 has color c and c, we have c = c

I Thus, h1, h2, . . . , hk+1 also have same color

I Whats the fallacy?


Recursive Definitions

I Should be familiar with recursive functions from programming:

public int fact(int n) {

if(n

Recursively Defined Functions

I Just like sequences, functions can also be defined recursively

I Example:

f (0) = 3f (n + 1) = 2f (n) + 3 (n 1)

I What is f (1)?

I What is f (2)?

I What is f (3)?


Recursive Definition Examples

I Consider f (n) = 2n + 1 where n is non-negative integer

I Whats a recursive definition for f ?

I Consider the sequence 1, 4, 9, 16, . . .

I What is a recursive definition for this sequence?

I Recursive definition of function defined as f (n) =n

i=1i?


Recursive Definitions of Important Functions

I Some important functions/sequences defined recursively

I Factorial function:

f (1) = 1f (n) = n f (n 1) (n 2)

I Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, . . .

a1 = 1a2 = 1an = an1 + an2 (n 3)

I Just like there can be multiple bases cases in inductive proofs,there can be multiple base cases in recursive definitions


Inductive Proofs for Recursively Defined Structures

I Recursive definitions and inductive proofs are very similar

I Natural to use induction to prove properties about recursivelydefined structures (sequences, functions etc.)

I Consider the recursive definition:

f (0) = 1f (n) = f (n 1) + 2

I Prove that f (n) = 2n + 1


Example

I Let fn denote the nth element of the Fibonacci sequence

I Prove: For n 3, fn > n2 where = 1+5

2

I Proof is by strong induction on n with two base cases

I Intuition 1: Definition of fn has two base cases

I Intuition 2: Recursive step uses fn1, fn2 strong induction

I Base case 1 (n=3): f3 = 2, and < 2, thus f3 >

I Base case 2 (n=4): f4 = 3 and 2 =(3+5)

2 < 3


Example, cont.

Prove: For n 3, fn > n2 where = 1+5

2

I Inductive step: Assuming property holds for fi where3 i k , need to show fk+1 > k1

I First, rewrite k1 as 2k3

I 2 is equal to 1 + because:

2 =

(1 +

5

2

)2=

5 + 3

2= + 1

I Thus, k1 = (+ 1)(k3) = k2 + k3


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Example, cont.

I k1 = k2 + k3

I By recursive definition, we know fk+1 = fk + fk1

I Furthermore, by inductive hypothesis:

fk > k2 fk1 > k3

I Therefore, fk+1 > k2 + k3 = k1


Recursively Defined Sets and Structures

I We saw how to define functions and sequences recursively

I We can also define sets and other data structures recursively

I Example: Consider the set S defined as:

3 SIf x S and y S , then x + y S

I What is the set S defined as above?


More Examples

I Give a recursive definition of the set E of all even integers:

I Base case:

I Recursive step:

I Give a recursive definition of N, the set of all natural numbers:

I Base case:

I Recursive step:


Strings and Alphabets

I Recursive definitions play important role in study of strings

I Strings are defined over an alphabet

I Example: 1 = {a, b}

I Example: 2 = {0}

I Examples of strings over 1: a, b, aa, ab, ba, bb, . . .

I Set of all strings formed from forms language called

I 2 = {, 0, 00, 000, . . .}


Recursive Definition of Strings

I The language has natural recursive definition:

I Base case: (empty string)

I Recursive step: If w and x , then wx

I Since is the empty string, s = s

I Consider the alphabet = {0, 1}

I How is the string 1 formed according to this definition?

I How is 10 formed?


Recursive Definitions of String Operations

I Many operations on strings can be defined recursively.

I Consider function l(w) which yields length of string w

I Example: Give recursive definition of l(w)

I Base case:

I Recursive step:


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Another Example

I The reverse of a string s is s written backwards.

I Example: Reverse of abc is bca

I Give a recursive definition of the reverse(s) operation

I Base case:

I Recursive step:


Palindromes

I A palindrome is a string that reads the same forwards andbackwards

I Examples: mom, dad, abba, Madam Im Adam, . . .

I Give a recursive definition of the set P of all palindromes overthe alphabet = {a, b}

I Base cases:

I Recursive step:


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