# Relations and Functions-II, Objectives, Relation, Types of Relations

Doorsteptutor material for IAS is prepared by world's top subject experts: Get detailed illustrated notes covering entire syllabus: point-by-point for high retention.

Download PDF of This Page (Size: 160K) ↧

## Objectives

After studying this lesson, you will be able to:

## Relation

Let and be two sets. Then a relation from Set into Set is a subset of .

Thus, is a relation from to

If then we write which is read as ‘’ is related to by the relation , if , then we write and we say that is not related to by the relation .

If and , then has mn ordered pairs, therefore, total number of relations form to is .

### Types of Relations

**(i) Reflexive Relation:**

A relation on a set is said to be reflexive if every element of is related to itself.

Thus, is reflexive for all

A relation is not reflexive if there exists an element such that .

Let be a set. Then

is a reflexive relation on .

But is not a reflexive relation on , because but .

**(ii) Symmetric Relation:**

A relation on a set is said to be symmetric relation if

for all

i.e. for all .

Let and and be relations on given by

and

is symmetric relation on because

or for all

but is not symmetric because but .

A reflexive relation on a set is not necessarily symmetric. For example, the relation

is a reflexive relation on set but it is not symmetric.

**(iii) Transitive Relation:**

Let be any set. relation on is said to be transitive relation if

and for all

i.e. and for all

For example:

On the set of natural numbers, the relation defined by

‘ is less than ’, is transitive, because for any

and

i.e. and

Example:

Check the relation for reflexivity, symmetry and transitivity, where is defined as iff for all

Solution:

Let be the set of all lines in a plane. Given that for all

Reflexivity: is not reflexive because a line cannot be perpendicular to itself i.e. is not true.

Symmetry: Let such that

Then

So, is symmetric on

Transitive: is not transitive, because and does not impty that