Review: Induction and Strong Induction. CS311H: Discrete Mathematics. Mathematical Induction. Matchstick Example. Example. Matchstick Proof, cont.


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1 Review: Induction and Strong Induction CS311H: Discrete Mathematics Mathematical Induction Işıl Dillig How does one prove something by induction? What is the inductive hypothesis? What is the difference between regular and strong induction? Is strong induction more powerful than standard induction? Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 1/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction /7 Example Matchstick Example For n 1, prove there exist natural numbers a, b such that: Hint: 5 n+1 = 5 5 n 1 5 n = a + b The Matchstick game: There are two piles with same number of matches initially Two players take turns removing any positive number of matches from one of the two piles Player who removes the last match wins the game Prove: Second player always has a winning strategy. Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 3/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 4/7 Matchstick Proof Matchstick Proof, cont. P(n): Player has winning strategy if initially n matches in each pile Induction: Assume j.1 j k P(j ); show P(k + 1) Inductive hypothesis: Prove Player wins if each pile contains k + 1 matches Case 1: Player 1 takes k + 1 matches from one of the piles. What is winning strategy for player? Case : Player 1 takes r matches from one pile, where 1 r k Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 5/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 6/7 1
2 Recursive Definitions Recursive Definitions in Math Should be familiar with recursive functions from programming: public int fact(int n) { if(n <= 1) return 1; return n * fact(n  1); } Recursive definitions are also used in math for defining sets, functions, sequences etc. Consider the following sequence: 1, 3, 9, 7, 81,... This sequence can be defined recursively as follows: a 0 = 1 a n = 3 a n 1 First part called base case; second part called recursive step Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 7/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 8/7 Recursively Defined Functions Recursive Definition Examples Just like sequences, functions can also be defined recursively Example: What is f (1)? What is f ()? What is f (3)? f (0) = 3 f (n + 1) = f (n) + 3 (n 1) Consider f (n) = n + 1 where n is nonnegative integer What s a recursive definition for f? Consider the sequence 1, 4, 9, 16,... What is a recursive definition for this sequence? Recursive definition of function defined as f (n) = n i? i=1 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 9/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 10/7 Recursive Definitions of Important Functions Inductive Proofs for Recursively Defined Structures Some important functions/sequences defined recursively Factorial function: f (1) = 1 f (n) = n f (n 1) (n ) Fibonacci numbers: 1, 1,, 3, 5, 8, 13, 1,... a 1 = 1 a = 1 a n = a n 1 + a n (n 3) Just like there can be multiple bases cases in inductive proofs, there can be multiple base cases in recursive definitions Recursive definitions and inductive proofs are very similar Natural to use induction to prove properties about recursively defined structures (sequences, functions etc.) Consider the recursive definition: Prove that f (n) = n + 1 f (0) = 1 f (n) = f (n 1) + Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 11/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 1/7
3 Example Let f n denote the n th element of the Fibonacci sequence Prove: For n 3, f n > α n where α = 1+ 5 Proof is by strong induction on n with two base cases Intuition 1: Definition of f n has two base cases Intuition : Recursive step uses f n 1, f n strong induction Base case 1 (n=3): f 3 =, and α <, thus f 3 > α Base case (n=4): f 4 = 3 and α = (3+ 5) < 3 Example, cont. Prove: For n 3, f n > α n where α = 1+ 5 Inductive step: Assuming property holds for f i where 3 i k, need to show f k+1 > α k 1 First, rewrite α k 1 as α α k 3 α is equal to 1 + α because: α = ( 1 + ) = = α + 1 Thus, α k 1 = (α + 1)(α k 3 ) = α k + α k 3 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 13/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 14/7 Example, cont. Recursively Defined Sets and Structures We can also define sets and other data structures recursively α k 1 = α k + α k 3 By recursive definition, we know f k+1 = f k + f k 1 Furthermore, by inductive hypothesis: Example: Consider the set S defined as: 3 S If x S and y S, then x + y S f k > α k f k 1 > α k 3 What is the set S defined as above? Therefore, f k+1 > α k + α k 3 = α k 1 Give a recursive definition of the set E of all even integers: Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 15/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 16/7 Strings and Alphabets Recursive Definition of Strings The language Σ has natural recursive definition: Recursive definitions play important role in study of strings Strings are defined over an alphabet Σ ɛ Σ (empty string) If w Σ and x Σ, then wx Σ Example: Σ1 = {a, b} Set of all strings formed from Σ forms language called Σ Σ 1: {ɛ, a, b, aa, ab, ba, bb,...} Since ɛ is the empty string, ɛs = s Consider the alphabet Σ = {0, 1} How is the string 1 formed according to this definition? How is 10 formed? Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 17/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 18/7 3
4 Recursive Definitions of String Operations Another Example Many operations on strings can be defined recursively. Consider function l(w) which yields length of string w Example: Give recursive definition of l(w) The reverse of a string s is s written backwards. Example: Reverse of abc is bca Give a recursive definition of the reverse(s) operation Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 19/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 0/7 Palindromes Structural Induction A palindrome is a string that reads the same forwards and backwards Examples: mom, dad, abba, Madam I m Adam,... Give a recursive definition of the set P of all palindromes over the alphabet Σ = {a, b} Base cases: How do we prove properties about recursively defined structures? Stuctural induction is a technique that allows us to apply induction on recursive definitions even if there is no integer Structural induction is also no more powerful than regular induction, but can make proofs much easier Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 1/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction /7 Structural Induction Overview Suppose we have: a recursively defined structure S a property P we d like to prove about S Structural induction works as follows: 1. Prove P about base case in recursive definition. Inductive step: Assuming P holds for substructures used in the recursive step of the definition, show that P holds for the recursively constructed structure. Example 1 Consider the following recursively defined set S: 1. a S. If x S, then (x) S Prove by structural induction that every element in S contains an equal number of right and left parantheses. a has 0 left and 0 right parantheses Inductive step: By the inductive hypothesis, x has equal number, say n, of right and left parantheses. Thus, (x) has n + 1 left and n + 1 right parantheses. Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 3/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 4/7 4
5 Example Proof, Part I Consider the set S defined recursively as follows: 3 S Consider the set S defined recursively as follows: 3 S and if x S and y S, then x + y S If x S and y S, then x + y S Let s first prove S A, i.e., any element in S is divisible by 3 Prove S is set of all positive integers that are multiples of 3 Let A be the set of all positive integers divisble by 3 We want to show that A = S For this, we ll use structural induction Inductive step: To do this, we need to prove S A and A S Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 5/7 Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 6/7 Proof, Part II We showed that all integers in S are multiples of 3, but still need to show S includes all positive multiples of 3 Therefore, need to prove that 3n S for all n 1 We ll prove this by strong induction on n: Base case (n=1): Inductive hypothesis: Need to show: Işıl Dillig, CS311H: Discrete Mathematics Mathematical Induction 7/7 5
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