Functions

Introduction

Definition 1: A function or mapping f from a set A to a set B, denoted f : A ? B, is a correspondence in which to each element x of A corresponds exactly one element y = f(x) of B.

Definition 2. Let A and B be sets. A function f with type A ? B is a relation with domain A and codomain B, such that ?x ? A. ?y1 ? B. ?y2 ? B. ((x, y1) ? f ? (x, y2) ? f ) ? y1 = y2.

Examples on Functions

The function Floor accepts any real number as input and outputs the integer formed by truncating the fractional part of the number input. For example, Floor(3.14159) = ?3.14159? = 3.
The function Ceiling accepts any real number as input and outputs the smallest integer greater than or equal to the number input. For example, Ceiling(3.14159) = ?3.141591? = 4. This function is also referred to as the greatest integer function.

Domain and Range

The domain of a function is the set of all things that may be input to produce some output.
The range of a function is the set of all things that are output. Once one knows the domain of a function, one can determine the range by applying the function to each element of the domain.
The codomain of a function is the set of all values that are potential outputs.

Identity Function

Definition: Given a set A, the function 1A : A ? A defined by 1A(x) = x for every x in A is called the identity function for A.

Example
Let A={1,2,3} and f is the identity function, then f(1) =1, f(2)=2, and f(3)=3.