# Concept of Functions

Let A and B be any two non–empty sets. Then a function ‘’ is a rule or law which associates each element of ‘A’ to a unique element of set ‘B’.

__Notation:__

(i) A function is usually denoted by small letters, i.e. etc. and Greek letters, i.e. etc.

(ii) If ‘’ is a function from ‘A’ to ‘B’ then we write .

__Ordered Pair:__

Let ‘a’ and ‘b’ be any two elements then an element is called an ordered pair.

__Cartesian product:__

e.g. ,

Let ‘A’ and ‘B’ be any two non–empty sets, then the set of all those elements of the form, where , is called a Cartesian product.

It is denoted by

__Binary Relation:__

Let ‘A’ and ‘B’ be any two non–empty sets, then every sub–set of is called a binary relation from A to B.

It is denoted by ‘’i.e.

e.g. ,

‘’ is a binary relation from A to B.

__Function as a Binary Relation:__

Let ‘A’ and ‘B’ be any two non–empty sets, then a binary relation ‘’ from ‘A’ to ‘B’ is called a function if it satisfied the following two conditions.

(i) Domain of , i.e.

(ii) For each element ‘x’ of ‘A’ there exists a unique element such that

__Domain of a Relation:__

The set of the first elements of all ordered pairs in a relation is called the domain of a relation.

e.g.

Dom

__Range of a Relation:__

The set of the second elements of the ordered pair in a relation is called the range of a relation.

e.g.

Range

__Domain of the Function:__

e.g.

For

For

For

Domain

For the set of all values of ‘x’ for which ‘y’ is finite, the definite real number is called the domain of the function.