# DOMAIN AND RANGE OF A RELATION

Domain and Range of a Relation :

If R is a relation from A to B, then the set of all first co-ordinates of elements of R is called the domain of R, while the set of all second co-ordinates of elements of R is called the range of R.

So,

Domain (R)  =  {a : (a, b) ∈ R}

Range (R)  =  {b : (a, b) ∈ R}

## Domain and Range of a Relation - Example

Let A  =  {1, 2, 3} and B  =  {2, 4}

Then,

A x B  =  {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}

By definition, every subset of A x B is a relation from A to B.

Thus, is we consider the relation

R  =  {(1, 2), (1, 4), (3, 2), (3, 4)}

then,

Domain (R)  =  {1, 3}

Range (R)  =  {2, 4}

## Domain and Range of a Relation - Practice Problems

Problem 1 :

Let A  =  {1, 2, 3, 4} and B  =  {1, 4, 9, 16, 25}. If R is the relation which maps the elements from A to B using the rule f(x)  =  x2, then find the domain and range of R.

Solution :

R maps the elements from A to B using the rule

f(x)  =  x2

Then, we have

f(1)  =  12  =  1

f(2)  =  22  =  4

f(3)  =  32  =  9

f(4)  =  42  =  16

So,

R  =  {(1, 1), (2, 4), (3, 9), (4, 16)}

Therefore,

Domain (R)  =  {1, 2, 3, 4}

Range (R)  =  {1, 4, 9, 16}

Problem 2 :

Let A  =  {1, 2, 3} and B  =  {5, 6, 7, 8}.

R is the relation which maps the elements from A to B as shown below.

Find the domain and range of R.

Solution :

From the arrow diagram shown above,

R  =  {(1, 5), (2, 8), (3, 6)}

Therefore,

Domain (R)  =  {1, 2, 3}

Range (R)  =  {5, 6, 8}

Problem 3 :

Let X  =  {a, b, c} and Y  =  {d, e, f, g}.

R is the relation which maps the elements from X to Y as shown below.

Find the domain and range of R.

Solution :

From the arrow diagram shown above,

R  =  {(a, e), (c, e), (d, f)}

Therefore,

Domain (R)  =  {a, c, d}

Range (R)  =  {d, e}

Problem 4 :

Let R be a relation defined as given below.

R  =  {(1, 1), (2, 3), (3, 4), (2, 7)}

Find the domain and range of R and R-1. Discuss the relationship between the domain and range of R and R-1

Solution :

R  =  {(1, 1), (2, 3), (3, 4), (2, 7)}

Domain and range of R :

Domain (R)  =  {1, 2, 3}

Range (R)  =  {1, 3, 4, 7}

Find inverse relation R-1 :

R-1  =  {(1, 1), (3, 2), (4, 3), (7, 2)}

Domain and range of R-1 :

Domain (R-1)  =  {1, 3, 4, 7}

Range (R-1)  =  {1, 2, 3}

Clearly,

Domain (R-1)  =  Range (R)

Range (R-1)  =  Domain (R)

After having gone through the stuff given above, we hope that the students would have understood domain and range of a relation.

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