Discrete Structures & Algorithms

More on Methods of Proof / Mathematical Induction

EECE 320 — UBC

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Announcements

First assignment due Friday• drop-box

Quiz during next lecture

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Quiz sample

1.) Show that the following statement: For every positive integer n,

2n n2

is false

2.) Show that if x is irrational, then sqrt(x) is irrational.

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What have we explored so far?

Propositional logicRules of inferenceMethods of proofProofs by induction

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The essential elements of mathematical induction are:

•The proposition of interest

•The base case•The inductive step

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Example

Prove that 2 n < n! for all n > 3

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Example

Prove that n3-n is divisible by 3 for all integers n

Do you need to use ‘strong induction’?

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ExampleConsider 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

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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, 0j k.

Aim to show that 5(k+1)=0.

We can write k+1=i+j with 0i,j k. 5(k+1)=5(i+j)=5i+5j=0+0=0.

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Recursive definitions

Series: Sn = 1 + 2 + 3 + … + n

Can be defined as:S1 = 1

Sn+1 = Sn + (n+1)

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Recursive definitions

Sets: S3 = {n| n>0, n divisible by 3}

Can recursively be defined as:Base step: 3 S3

Recursive step: if x S3 and y S3 then x+y S3

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Recursive definitions

Structures: A list Ln …is formed by its cells

C1, C2 ,,, Cn

A list recursively be defined as:Base step: an empty list is a list. Recursive step: Ln+1 is a list if it is formed by adding a cell C to a list Ln

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Recursive algorithms

Naturally work on all recursive definitions!

Example: Insertion sort

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To insert 12, we need to make room for it by moving first 36 and then 24.6 10 24

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36

Insertion sort

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6 10 24 36

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Insertion sort

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6 10 24 36

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Insertion sort

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Insertion sort

Procedure sort (List L):if length(L) 1

return Lelse

C = head (L)T = tail (L)return insert (C, sort(T))

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(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))

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Induction proofs and recursive procedures

You can use proofs based on induction to prove:• Correctness invariants • Complexity limits

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Insertion sort – correctness

Procedure sort (List L):if length(L) 1

return Lelse

C = head (L)T = tail (L)return insert (C, sort(T))

L1 sorted

if LN

sorted then LN+1 sortedTo prove: LN sorted for all lengths N

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Insertion sort – complexity

Procedure sort (List L):if length(L) 1

return Lelse

C = head (L)T = tail (L)return insert (C, sort(T))

P1 = 0

PN+1 = PN + complexity(insertN)

To find: PN – the average number of operations to sort LN

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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

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Better sorting

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Procedure mergesort (L=a1, a2, ….an):

if n>1 then

m:= n/2

L1 = a1, a2, …. am

L2 = am+1, …. an L = merge (mergesort(L1),mergesort(L2))

return L

Recursive proof Correctness Complexity

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The essential elements of mathematical induction are:

•The proposition of interest

•The base case•The inductive step

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Various uses of induction•Normal•Strong induction

Recursive definitions •Induction: To prove correctness properties (invariants)

• Induction: To analyze algorithm complexity

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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.