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

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## 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

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