Discrete Structures & Algorithms More on Methods of Proof / Mathematical Induction EECE 320 — UBC

2 Announcements First assignment due Friday drop-box Quiz during next lecture

3 Quiz sample 1.) Show that the following statement: For every positive integer n, 2n  n 2 is false 2.) Show that if x is irrational, then sqrt(x) is irrational.

4 What have we explored so far? Propositional logic Rules of inference Methods of proof Proofs by induction

5 The essential elements of mathematical induction are: The proposition of interest The base case The inductive step

6 Example Prove that 2 n 3

7 Example Prove that n 3 -n is divisible by 3 for all integers n Do you need to use ‘strong induction’?

8 Example Consider the following game: Two players. Players take turns removing any positive number of matches from one of two piles of matches. The player who takes the last match(es) wins. Show that if the two piles contain the same number of matches, the second player has a guaranteed winning strategy

9 What’s wrong with the following proof using ‘strong induction’? For all non-negative integer n, 5n=0 Proof: Base case: 5*0=0 Induction step: Suppose that 5j = 0 for all non-negative integers j, 0  j  k. Aim to show that 5(k+1)=0. We can write k+1=i+j with 0  i,j  k. 5(k+1)=5(i+j)=5i+5j=0+0=0.

10 Recursive definitions Series: S n = … + n Can be defined as: S 1 = 1 S n+1 = S n + (n+1)

11 Recursive definitions Sets: S 3 = {n| n>0, n divisible by 3} Can recursively be defined as: Base step: 3  S 3 Recursive step: if x  S 3 and y  S 3 then x+y  S 3

12 Recursive definitions Structures: A list L n …is formed by its cells C 1, C 2,,, C n A list recursively be defined as: Base step: an empty list is a list. Recursive step: L n+1 is a list if it is formed by adding a cell C to a list L n

13 Recursive algorithms Naturally work on all recursive definitions! Example: Insertion sort

14 To insert 12, we need to make room for it by moving first 36 and then Insertion sort

Insertion sort

Insertion sort

17 Insertion sort Procedure sort (List L): if length(L)  1 return L else C = head (L) T = tail (L) return insert (C, sort(T))

18 (auxiliary procedure: Insertion) Procedure insert (Cell C, List L): if length(L) = 0 return list(C) else if C < head (L) return list(C,L) else return list(head(L), insert (C, tail(L))

19 Induction proofs and recursive procedures You can use proofs based on induction to prove: Correctness invariants Complexity limits

20 Insertion sort – correctness Procedure sort (List L): if length(L)  1 return L else C = head (L) T = tail (L) return insert (C, sort(T)) L 1 sorted if L N sorted then L N+1 sorted To prove: L N sorted for all lengths N

21 Insertion sort – complexity Procedure sort (List L): if length(L)  1 return L else C = head (L) T = tail (L) return insert (C, sort(T)) P 1 = 0 P N+1 = P N + complexit y(insert N ) To find: P N – the average number of operations to sort L N

22 In class exercise Give a recursive algorithm to find the maximum of a list of non-negative integers. Use structural induction to prove correctness Analyze complexity

23 Better sorting

24 Procedure mergesort (L=a 1, a 2, ….a n ): if n>1 then m:= n/2 L 1 = a 1, a 2, …. a m L 2 = a m+1, …. a n L = merge (mergesort(L 1 ),mergesort(L 2 )) return L Recursive proof  Correctness  Complexity

25 The essential elements of mathematical induction are: The proposition of interest The base case The inductive step

26 Various uses of induction Normal Strong induction Recursive definitions Induction: To prove correctness properties (invariants) Induction: To analyze algorithm complexity

27 To understand how to perform computations efficiently, we need to know How to count! How to sum a sequence of steps. Reason about the complexity/efficiency of an algorithm. Prove the correctness of our analysis. –Logic –Methods of inference –Proof strategies –Induction –… (and eventually) How to create better algorithms.