# Domain and Range of a Relation

## What is domain of a Relation?

All the values that can go into the relation is called domain.

Imagine domain as input values which can be entered into the relation.

All the first element of the ordered pair in the set R is the domain of the relation.

For Example;
Consider the below relation set.
R = { (1, 5), (4, 7), (6, 10), (11, 13) }

The relation set consists of 4 ordered pairs.

The first element of all the ordered pairs is the domain of relation R.

Expressing the above relation in Arrow Diagram

Conclusion
Domain is the input value of the relation.
All the first element of the ordered pair is the domain value.

## What is the Range of the Relation?

When we put some input in the relation, we get an output.

All the possible output values of the relation is called Range.

All the second element in the ordered pair of set R is the Range of Relation.

Taking the same above example;
Given below is the relation set R with 4 ordered pairs.
R = { (1, 5), (4, 7), (6, 10), (11, 13) }

The second element of each ordered pairs is the Range of relation R.

Given below is the arrow diagram of above relation.

Conclusion
Range is the output value of relation.
The second value of ordered pair is part of the range.

## Solved Problems on Domain & Range

(01) Given below is the Relation Set R of student name & hair color.
Find the domain and range of relation R.

Solution
Note that there are 5 ordered pair in above relation.

The first part of the ordered pair is known as Domain.
Domain = { Black, Brown, Blonde, Red }

The second part of the ordered pair is called Range.
Range = { John, Tom, Jordan, Suzy, Jack }

Note that the domain are the input value and range are the output value of any relation.

So if you input Black, you will get John as output.

If you input Brown, you will get Tom & Jack as output.

(02) Given below are sets A, B and relation R.
A = {2, 4, 5 }
B = { 3, 5, 6, 8, 10 }
R ⟹ x fully divides y
Where, x ϵ A & y ϵ B

Find;
(i) Relation (R) in Roster form
(ii) Domain and Range of relation R

Solution
For set A & B, the relation states that x fully divides y.

Consider first element of set A ( i.e. 2 )
⟹ 2 divides element 6, 8, 10 of set B.
(2, 6), (2, 8), (2, 8) are the ordered pairs

Consider second element of set A (i.e. 4 )
⟹ 4 divides element 8 of set B
Here we get (4, 8) ordered pairs

Consider third element of set A ( i.e 5)
⟹ 5 divides element 5 & 10 of set B.
We get (5, 5) & (5, 10) as ordered pairs.

Combining all the ordered pairs, we will get the relation set R.
R = { (2, 6), (2, 8), (2, 8), (4, 8), (5, 5) (5, 10) }

(i) Domain of R
There are 6 ordered pair in above relation.

The first part of all ordered pair is the domain.

Hence, Domain = { 2, 4, 5 }

(ii) Range
The second part of all the ordered pair is the Range

Hence, Range = { 6, 8, 5, 10 }

(03) Given below is the arrow form of relation R.
(i) Find Domain and Range
(ii) Express Relation (R) in Roster form
(iii) Express Relation (R) in Set builder form

Solution
(i) Domain
All the input value of relation is called domain.
Domain = { 2, 4, 5, -3 }

Range
All the output value of relation is called Range
Range = { 10, 4, 8, 6 }

(ii) Express relation in Roster form
R = { (2, 4), (4, 8), (5, 10), (-3, 6)

(iii) Express relation in set builder form
R = { (a, b); b is twice of a & a ϵ A, b ϵ B }

(04) Find the domain and range of below relation.
R = { (2x, 3x) : x ϵ { 2, 3, 5, 7} }

Solution
Let’s find the value of ordered pair.

Put x = 2 in (2x, 3x)
We get ordered pair ( 4, 6 )

Put x = 3 in ( 2x, 3x )
We get ordered pair ( 6, 9 )

Put x = 5
we get ordered pair ( 10, 15 )

Put x = 7
we get ordered pair ( 14, 21 )

Collecting all the ordered pair, we get the set R.
R = { (4, 6), (6, 9), (10, 15), (14, 21) }

Domain
All the first element of the ordered pair are the domains.
Domain = { 4, 6, 10, 14 }

Range
All the second element of ordered pair are the Range
Range = { 6, 9, 15, 21 }