Announcements CS311H: Discrete Mathematics Mathematical Induction Homework 5 out today

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Announcements
CS311H: Discrete Mathematics
Mathematical Induction
I¸sıl Dillig
I¸sıl Dillig,
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Homework 5 out today
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Due next Thursday (Oct 30)
I¸sıl Dillig,
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Review: Strong Induction
Example
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Prove that every integer n ≥ 12 can be written as
n = 4a + 5b for some non-negative integers a, b.
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Base case same as regular induction, different in inductive step
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Proof by strong induction on n and consider 4 base cases
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Regular induction: assume P (k ) holds and prove P (k + 1)
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Base case 1 (n=12): 12 = 3 · 4 + 0 · 5
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Strong induction: assume P (1), P (2), .., P (k ); prove P (k + 1)
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Base case 2 (n=13): 13 = 2 · 4 + 1 · 5
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Base case 3 (n=14): 14 = 1 · 4 + 2 · 5
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Base case 4 (n=15): 15 = 0 · 4 + 3 · 5
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Example, cont.
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Matchstick Example
Prove that every integer n ≥ 12 can be written as n = 4a + 5b for
some non-negative integers a, b.
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Inductive hypothesis: Suppose every 12 ≤ i ≤ k can be
written as i = 4a + 5b.
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Inductive step: We want to show k + 1 can also be written
this way for k + 1 ≥ 16
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Observe: k + 1 = (k − 3) + 4
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By the inductive hypothesis, k − 3 = 4a + 5b for some a, b
because k − 3 ≥ 12
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But then, k + 1 can be written as 4(a + 1) + 5b
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1
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The Matchstick game: There are two piles with same number
of matches initially
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Two players take turns removing any positive number of
matches from one of the two piles
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Player who removes the last match wins the game
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Prove: Second player always has a winning strategy.
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Matchstick Proof
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Matchstick Proof, cont.
P (n): Player 2 has winning strategy if initially n matches in
each pile
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Base case:
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Induction: Assume ∀j .1 ≤ j ≤ k → P (j ); show P (k + 1)
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Inductive hypothesis:
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Prove Player 2 wins if each pile contains k + 1 matches
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Case 1: Player 1 takes k + 1 matches from one of the piles.
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What is winning strategy for player 2
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Case 2: Player 1 takes r matches from one pile, where
1≤r ≤k
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Now, player 2 takes r matches from other pile
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Now, the inductive hypothesis applies ⇒ player 2 has winning
strategy for rest of the game
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The Horse Paradox
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Induction: Assume P (k ); prove P (k + 1)
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Consider a collection of k + 1 horses: h1 , h2 , . . . , hk +1
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Consider the statement: All horses have the same color
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By IH, h1 , h2 , . . . , hk have the same color; let this color be c
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What is wrong with the following bogus proof of this
statement?
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By IH, h2 , . . . , hk +1 have same color; call this color c 0
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P (n) : A collection of n horses have the same color
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Since h2 has color c and c 0 , we have c = c 0
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Base case: P (1) X
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Thus, h1 , h2 , . . . , hk +1 also have same color
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What’s the fallacy?
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Recursive Definitions
Should be familiar with recursive functions from programming:
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This sequence can be defined recursively as follows:
a0
an
Recursive definitions are also used in math for defining sets,
functions, sequences etc.
Consider the following sequence:
1, 3, 9, 27, 81, . . .
public int fact(int n) {
if(n <= 1) return 1;
return n * fact(n - 1);
}
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Recursive Definitions in Math
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Bogus Proof, cont.
Easy to make subtle errors when trying to prove things by
induction – pay attention to details!
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2
= 1
= 3 · an−1
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First part called base case; second part called recursive step
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Very similar to induction; in fact, recursive definitions
sometimes also called inductive definitions
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Recursively Defined Functions
Recursive Definition Examples
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Just like sequences, functions can also be defined recursively
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Example:
f (0)
= 3
f (n + 1) = 2f (n) + 3 (n ≥ 1)
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Consider f (n) = 2n + 1 where n is non-negative integer
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What’s a recursive definition for f ?
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Consider the sequence 1, 4, 9, 16, . . .
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What is f (1)?
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What is a recursive definition for this sequence?
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What is f (2)?
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Recursive definition of function defined as f (n) =
n
P
i?
i=1
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What is f (3)?
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I¸sıl Dillig,
Recursive Definitions of Important Functions
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Some important functions/sequences defined recursively
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Factorial function:
Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, . . .
a1
a2
an
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= 1
= 1
= an−1 + an−2
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Recursive definitions and inductive proofs are very similar
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Natural to use induction to prove properties about recursively
defined structures (sequences, functions etc.)
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Consider the recursive definition:
f (0) = 1
f (n) = f (n − 1) + 2
(n ≥ 3)
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Prove that f (n) = 2n + 1
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Example
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Example, cont.
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Let fn denote the n’th element of the Fibonacci sequence
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Prove: For n ≥ 3, fn > αn−2 where α =
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Proof is by strong induction on n with two base cases
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Just like there can be multiple bases cases in inductive proofs,
there can be multiple base cases in recursive definitions
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Inductive Proofs for Recursively Defined Structures
f (1) = 1
f (n) = n · f (n − 1) (n ≥ 2)
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Prove: For n ≥ 3, fn > αn−2 where α =
√
1+ 5
2
Intuition 1: Definition of fn has two base cases
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Inductive step: Assuming property holds for fi where
3 ≤ i ≤ k , need to show fk +1 > αk −1
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First, rewrite αk −1 as α2 αk −3
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α2 is equal to 1 + α because:
Intuition 2: Recursive step uses fn−1 , fn−2 ⇒ strong induction
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Base case 1 (n=3): f3 = 2, and α < 2, thus f3 > α
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Base case 2 (n=4): f4 = 3 and α2 =
√
(3+ 5)
2
2
α =
<3
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3
√
1+ 5
2
√ !2 √
1+ 5
5+3
=
=α+1
2
2
Thus, αk −1 = (α + 1)(αk −3 ) = αk −2 + αk −3
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Example, cont.
Recursively Defined Sets and Structures
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αk −1 = αk −2 + αk −3
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We saw how to define functions and sequences recursively
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By recursive definition, we know fk +1 = fk + fk −1
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We can also define sets and other data structures recursively
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Furthermore, by inductive hypothesis:
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Example: Consider the set S defined as:
fk > αk −2
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fk −1 > αk −3
3∈S
If x ∈ S and y ∈ S , then x + y ∈ S
Therefore, fk +1 > αk −2 + αk −3 = αk −1
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I¸sıl Dillig,
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I¸sıl Dillig,
More Examples
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Base case:
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Recursive step:
Give a recursive definition of N, the set of all natural numbers:
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Base case:
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Recursive step:
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Recursive definitions play important role in study of strings
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Strings are defined over an alphabet Σ
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Example: Σ1 = {a, b}
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Example: Σ2 = {0}
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Examples of strings over Σ1 : a, b, aa, ab, ba, bb, . . .
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Set of all strings formed from Σ forms language called Σ∗
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Σ∗2 = {, 0, 00, 000, . . .}
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Recursive Definition of Strings
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Strings and Alphabets
Give a recursive definition of the set E of all even integers:
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What is the set S defined as above?
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Recursive Definitions of String Operations
The language Σ∗ has natural recursive definition:
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Base case: ∈ Σ∗ (empty string)
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Many operations on strings can be defined recursively.
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Recursive step: If w ∈ Σ∗ and x ∈ Σ, then wx ∈ Σ∗
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Consider function l (w ) which yields length of string w
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Example: Give recursive definition of l (w )
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Since is the empty string, s = s
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Consider the alphabet Σ = {0, 1}
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How is the string ”1” formed according to this definition?
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How is ”10” formed?
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4
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Base case:
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Recursive step:
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Another Example
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Palindromes
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A palindrome is a string that reads the same forwards and
backwards
Example: Reverse of ”abc” is ”bca”
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Examples: ”mom”, ”dad”, ”abba”, ”Madam I’m Adam”, . . .
Give a recursive definition of the reverse(s) operation
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Give a recursive definition of the set P of all palindromes over
the alphabet Σ = {a, b}
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Base cases:
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Recursive step:
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The reverse of a string s is s written backwards.
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Base case:
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Recursive step:
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5
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Announcements CS311H: Discrete Mathematics Mathematical Induction Homework 5 out today

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