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

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 n = n(n + 1 )/2 for all positive integers n.

Example

Recursion

Recursively Defined Functions  Simplest case: One way to define a function f:N  S (for any set S) or series a n =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 n  N recursively by: Let f(0)=3 For all n  N, 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) :  1 Recursive part: F(n) :  n  F(n-1). F(1)=1 F(2)=2 F(3)=6

The Fibonacci Series  The Fibonacci series f n  0 is a famous series defined by: f 0 :  0, f 1 :  1, f n  2 :  f n − 1 + f n −

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

Example

Recursive algorithm Factorial Algorithm procedure factorial( n: nonnegative integer) if n = 0 then factorial(n) : = 1 else factorial(n) := n ·factorial(n - 1 )

Example Algorithm to calculate a n procedure power( a : nonzero real number, n : nonnegative integer ) if n = 0 then power(a, n) : = 1 else power(a, n) := a · power(a, n - 1)

Example GCD Algorithm procedure gcd( a, b: nonnegative integers with a < b ) if a = 0 then gcd(a, b) := b else gcd(a, b) := gcd(b mod a, a)

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