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 Aร—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 Aร—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)โˆˆR, for all xโˆˆA. 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)โˆˆR โˆˆ (y,x)โˆˆR, for all x,y โˆˆA.

(iii) Transitive Relation

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

(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.

injective function

(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 yโˆˆB there exists some xโˆˆA such that y = f (x). Thus f is onto if and only if co-domain of f = Range of f.

subjective function

(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.

Bijective function