Domain and Range of a Relation

Before learning about Domain and Range of a Relation firstly know what Relations are. A relation is a rule that relates an element from one set to the other set. Consider two non-empty sets A and B then the relation is a subset of Cartesian Product AxB.

The domain is the set of all first elements of the ordered pairs. The range on the other hand is the set of all second elements of the ordered pairs. However, Range includes only the elements used by the function. There lies in a trick in the range i.e. Set B can be equal to the range of relation or bigger than that. This is because there can be elements in Set B that aren’t related to Set A.

Domain and Range Definition

If R be a relation from Set A to Set B then the Set of first elements belonging to the ordered pair is called the Domain of R. We can represent the Domain as such

Dom(R) = {a ∈ A: (a, b) ∈ R for some b ∈ B}

The Set of Second Components belonging to the ordered pair is called the Range of R. It can be denoted as follows

R = {b ∈ B: (a, b) ∈R for some a ∈ A}

Thus, Domain and Range are given by Domain (R) = {a : (a, b) ∈ R} and Range (R) = {b : (a, b) ∈ R}.

Solved Examples on Domain and Range of a Relation

1. State the domain and range of the following relation: (eye color, student’s name).
A = {(blue, John), (green, William), (brown,Wilson), (blue, Moy), (brown, Abraham), (green, Dutt)}. State whether the relation is a function?

Solution:

Domain: {blue, green, brown} Range: {John, William, Wilson, Moy, Abraham, Dutt}

No, the relation is not a function since the eye colors are repeated.

2. State the domain and range of the following relation: {(4,3), (-1,7), (2,-3), (7,5), (6,-2)}?

Solution:

The domain is the first component of the ordered pairs. Whereas, Range is the Second Component of the ordered pairs. Remove the duplicates if any are present.

Domain = {4, -1, 2, 7, 6} Range = {3, 7, -3, 5, -2}

3. From the following Arrow Diagram find the Domain and Range and depict the relation between them?

Solution:

Domain = {3, 4, 5}

Range = {3, 4, 5, 6}

R = {(3, 4), (4, 6), (5, 3), (5, 5)}

4. Determine the domain and range of the relation R defined by

R = {x – 2, 2x + 3} : x ∈ {0, 1, 2, 3, 4, 5}

Solution:

Given x = {0, 1, 2, 3, 4, 5}

x = 0 ⇒ x – 2 = 0 – 2 = -2 and 2x + 3 = 2*0 + 3 = 3

x = 1 ⇒ x-2 = 1-2 = -1 and 2x+3 = 2*1+3 = 5

x = 2 ⇒ x-2 = 2-2 = 0 and 2x+3 = 2*2+3 = 7

x = 3 ⇒ x-2 = 3-2 = 1 ad 2x+3 = 2*3+3 = 9

x = 4 ⇒ x-2 = 4-2 = 2 and 2x+3 = 2*4+3 = 11

x = 5 ⇒ x-2 = 5-2 =3 and 2x+3 = 2*5+3 = 13

Hence R = {-2, 3), (-1, 5), (0, 7), (1, 9), (2, 11), (3, 13)

Domain of R = {-2, -1, 0, 1, 2, 3}

Range of R = {3, 5, 7, 9, 11, 13}

5. The below figure shows a relation between Set x and Set y. Write the same in Roster Form, Set Builder Form, and find the domain and Range?

Solution:

In the Set Builder Form R = {(x, y): x is the square of y, x ∈ X, y ∈ Y}

In Roster Form R = {(2, 1)(4, 2)}

Domain = {2, 4}

Range = {1, 2}

6. The Arrow Diagram Shows the Relation R from Set C to Set D. Write the relation R in Roster Form?

Solution:

We observe the relation R using the Arrow Diagram Above

From that Relation R in Roster Form = {(2,20) ; (2, 40) ; (4, 40) ; (3, 30)}