Recursively Defined Functions - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Discrete Mathematics

CS 2610

October 21, 2008

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Recursively Defined Functions

We are familiar with the use of formulae to define

functions. But it is also possible to define some

functions using recursion.

Example:

f(n) = n

2

for n ∈ Z

≥

or…

f(0) = 0

f(n+1) = f(n) + (2n + 1), for n ∈ Z

≥

4

Recursively Defined Sets

We can define sets in the same manner, by

establishing a basis and recursive step.

Example: S = { x | x is an odd, positive integer }

 Basis: 1 ∈ S  Rec. Step: if x ∈ S, then x + 2 ∈ S

5

Recursively Defined Languages

As many of you will learn in theory, a language in

Computer Science refers to some set of strings

generated from an alphabet using some set of

syntax rules.

Often times the alphabet is the set {0,1}

Example: L = { x | x is a bitstring }

 Basis: 0 ∈ S, 1 ∈ S  Rec. Step: if x ∈ S, then x0 ∈ S and x1 ∈ S (concatenation)

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Why Define Recursively?

Why bother to define sets, functions, or structures

recursively, when there are many other ways to

define them?

Recursive definitions can greatly simplify the

inductive proof process.

Example: Fibonacci proof  show f(1) + f(3) + … + f(2n-1) = f(2n) for Z +