Discrete Mathematics Lecture 17: Induction and Recursion, Slides of Discrete Mathematics

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CSci 2011

Discrete Mathematics

Lecture 17

Interesting Induction

Someone with zero hairs is bald.

Someone with one more hair than a bald person is bald.

.

. turn the inductive crank……

.

Therefore, someone with

1,000, 000 hairs is bald.

What’s wrong with this induction?

Induction Example

Induction Hypothesis

P(k) = ∑ki=1 2i-1= k^2

Inductive Steps P(k+1) = ∑k+1i=1 2i- = ki=1 2i-1 + (2 (k+1) -1) = k^2 + (2k +1) = (k+1) 2

Second induction example

Show the sum of the first n positive even integers is n^2 + n Rephrased: ∀ n P(n) where P(n) = ∑ni=1 2 i = n 2

• n

The three parts: Base case Inductive hypothesis Inductive step

More Examples

Prove thatt if h> -1, then 1+nh ≤ (1+h)n^ for all non-negative integer n.

Prove that n^2 ≡ 1 mod 8 for all odd integer n.

Strong induction

Weak mathematical induction assumes P( k ) is true, and uses that (and only that!) to show P( k +1) is true

Strong mathematical induction assumes P(1), P(2), …, P( k ) are all true, and uses that to show that P( k +1) is true. [P(1) ∧ P(2) ∧ p(3) ∧ … ∧ P(k) ] → P(k+1)

Strong induction example 1

Inductive step: Show that P( k +1) is true

There are two cases:  k +1 is prime It can then be written as the product of k +  k +1 is composite It can be written as the product of two composites, a and b, where 2 ≤ a ≤ b < k + By the inductive hypothesis, both P( a ) and P( b ) are true

Strong Induction Examples 2

Prove that every amount of 12 cents or more can be formed using just 4-cent and 5-sent stamps. Basis Step  12 = 3 * 4 cent stamp  13 = 2 * 4 + 1 * 5  14 = 1 * 4 + 2 * 5  15 = 3 * 5 Inductive step: Suppose P(j) is true for 12 ≤ j ≤ k (k≥15). It is sufficient to show that P(k+1) is true. We know that P(k-3) is true since k ≥ 15. To form postage of k+1 cents, we just need to add 1 4-cent postage to the stamps we used to form k-3 cents.

Recursive objects

Sometimes it is difficult to define an object explicitly, but it is easier to define it in terms of itself.

Recursive Functions

We can define a function recursively by specifying: Basis: the value of the function at the smallest element of the domain. E.g.: f(0) = 1 Recursive step: A rule for finding the value of the function at an integer from its values at smaller integers E.g: f(n+1) = 2*f(n)

Many common functions can be defined recursively.

F(n) grows infinitely

Let α = (1+√5)/2 = 1.61803…

For n ≥3, F(n) > αn-

Proof by induction.

Hint for the inductive step:

Solve x 2 -x-1=0 for x

Strings can be defined recursively

Σ: Alphabet (set of symbols)

λ: Empty String

Σ*: set of all strings over the alphabet

Basis: λ∈Σ

Recursive Step: If w∈Σ* and x∈Σ, then wx ∈ Σ*

wx: string w followed by symbol x

(Rooted) Trees

• Basis: a single vertex r is a rooted tree
• Recursive step: Suppose that T 1 , …, Tn are rooted trees with roots r 1 , …, rn.
• Then, the graph formed by adding edges (r, ri) for i=1,…,n is a rooted tree

Extended Binary Trees

Basis: The empty set is an extended binary tree

Recursive step: if T 1 and T 2 are extended binary trees, then the tree T 1 T 2 obtained by connecting a root r to the roots of T 1 and T 2 is also an extended binary tree

r