# Relations.

**What is Relation?**

Any subset of a Cartesian product A×B in which the first element and second element of ordered pairs have special relation to each other is known as “Relation”.

A relation from one set (A) to another set (B) is denoted by:

“xRy” or simply “R”

Where (x,y)∈R

For example:

If set A={Me , My father , My son}

And set B={My spouse , My mother , My daughter}

Then **one of the “Relation”** from set A to set B can be:

R=[{My spouse , Me} , {My mother , My father}]

In above Relation , the relation between the first and second element of ordered pairs is that “first element is wife of Second element”. Like “My spouse” Is wife of “Me”

And

If set A={2,3,4}

And set B={4,5,6}

Then one of the “Relation” from set A to set B can be :

R=[{2,4} , {2,6} ,{3,6} ,{4,4}]

In above Relation the relation between first and second element of ordered pairs is that “First element is a factor of second element”. Like 4 is a factor of 4.

** Domain and Range of a Relation: **

“Domain” of a Relation(R) is the set of all the first elements of ordered pairs of the Relation(R)

and

“Range” of a Relation(R) is the set of all the second elements of ordered pairs of The Relation(R).

For Example:

If a Relation R=[{1,2} ,{2,3} ,{3,4}]

Domain of Relation R ={1,2,3}

And Range of Relation R={2,3,4}

Diagrammatically we can denote the relations from one set(A) to another (B) as following:

,

etc.

**Inverse Relations:**

A relation obtained by interchanging first and second elements in the ordered pairs of given Relation is known as the inverse Relation of given Relation.

If a Relation “R” is given then the inverse of the relation “R” is denoted by the symbol:

R^{-1}

For example:

If Relation R=[{1,2} , {3,4} , {5,6}}

Then the inverse of Relation R =R^{-1}=[{2,1} , {4,3} , {6,5}]

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