lecture induction2 6up
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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)
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 2/26
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)
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 3/26
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
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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
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 5/26
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
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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
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 8/26
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
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 9/26
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?
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 10/26
Recursive Definitions
I Should be familiar with recursive functions from programming:
public int fact(int n) {
if(n
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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)?
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 13/26
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?
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 14/26
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
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 15/26
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
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 16/26
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
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 17/26
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
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 18/26
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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
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 19/26
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?
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 20/26
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:
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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, . . .}
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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?
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 23/26
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:
Isl Dillig, CS311H: Discrete Mathematics Mathematical Induction 25/26
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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