## Chapter:1

# What are Relations and Functions ?

## Relation

Let A and B be two non-empty sets. A relation from set A to set B is a subset of A×B. For example: If A={2,4,6} and B={4,9,16,25,36}, then R={(2,4),(4,16),(6,36)} is a relation from A to B as R is a subset of A×B.

## Types Of Relation

One of the most important relation, which plays a significant role in Mathematics is an equivalence relation. To study equivalence relation, we first consider three types of relations, namely reflexive, symmetric and transitive.

**(i) Reflexive Relation**

A relation R in a given non-empty set A is called reflexive if (x,x)**∈**R, for all x**∈**A. Equivalently R is called reflexive in A if each element of A is related to itself.

**(ii) Symmetric Relation**

A relation R in a given non-empty set A is said to be symmetric (x,y)**∈**R **∈** (y,x)**∈**R, for all x,y **∈**A.

**(iii) Transitive Relation**

A relation R on a given non-empty set A is said to be transitive if (x,y)**∈**R and (y,z)**∈**R **∈** (x,z)**∈**R, for all x,y,z**∈**A.

**(iv) Equivalence Relation**

A relation R on a given non-empty set A is said to be an equivalence if it is reflexive, symmetric and transitive.

## Functions

Let A and B be two non-empty sets, then a relation f which associates each element of A with a unique element of B is called a function from A to B. It is also known as map or mapping.

## Types Of Functions

**(i) One to one or injective function**

A function f : A B is said to be one-one or an injective if each element of set A has

unique and different image in set B. Otherwise, f is called many-one.

**(ii) Onto function or subjective function**

A function f : A B is said to be an onto function if and only if each element of B is

the image of some element of A under f i.e., for every y**∈**B there exists some x**∈**A such that y = f (x). Thus f is onto if and only if co-domain of f = Range of f.

**(iii) One-one onto or Bijective function**

A function f : A B is said to be one-one onto if and only if it is both one-one and onto i.e., each element of set A has unique and different image in set B and each element of B has pre-image in A.