INDUCTION AND RECURSION

PRINCIPLE OF MATHEMATICAL INDUCTION

To prove that P(n) is true for all positive integers n , where P(n) is a propositional function, we complete two steps:

BASIS STEP:

We verify that P(1) is true.

INDUCTIVE STEP:

We show that the conditional statement P(k) P(k + 1 ) is true for all positive integers k.

Expressed as a rule of inference, this proof technique can be stated as

Example

Example

This last equation shows that P (k + 1) is true under the assumption that P(k) is true. This completes the inductive step.We have completed the basis step and the inductive step, so by mathematical induction we know that P (n) is true for all positive integers n. That is, we have proven that 1 + 2 + . . . + n = n(n + 1 )/2 for all positive integers n.

Example

Recursion

Recursively Defined Functions Simplest case: One way to define a

function f:NS (for any set S) or series an=f(n) is to: Define f(0). For n>0, define f(n) in terms of f(0),

…,f(n−1).

Another Example

Suppose we define f(n) for all nN recursively by:

Let f(0)=3For all nN, let f(n+1)=2f(n)+3

What are the values of the following?f(1)= 2f(0)+3=2.3+3=9

f(2)= 2f(1)+3=2.9+3=21 f(3)= 2f(2)+3=2.21+3=45 f(4)= 2f(3)+3=2.45+3=93

Recursive definition of Factorial

Give a recursive definition of the factorial function F(n) : n!

Base case: F(0) : 1Recursive part: F(n) : n F(n-1).

F(1)=1 F(2)=2 F(3)=6

The Fibonacci Series

The Fibonacci series fn0 is a famous series defined by:

f0 : 0, f1 : 1, fn2 : fn−1 + fn−2

0

1 1

2 3

5 8

13

Example

Give a recursive definition of an, where a is a nonzero real number and n is a nonnegative integer.

Example

Recursive algorithm

Factorial Algorithmprocedure factorial( n: nonnegative integer)if n = 0

then factorial(n) : = 1else

factorial(n) := n ·factorial(n - 1 )

Example

Algorithm to calculate an

procedure power(a : nonzero real number, n :

nonnegative integer)if n = 0

then power(a , n) : = 1else

power(a , n) := a · power(a , n - 1)

Example

GCD Algorithmprocedure gcd(a , b: nonnegative integers with a < b)if a = 0

then gcd(a , b) := belse

gcd(a , b) := gcd(b mod a , a)

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