To understand the relation, you need to have a clear idea of the cross-product of two sets. Because a relation is the subset of the cartesian product of two sets. Generally, a set is a collection of well-defined elements and a cartesian product is the product of two sets which has a set of ordered pairs in the form of (a, b). Where a is an element of the first set, b is an element of the second set. Basically, a relation is a rule that related an element from one set to the second element in another set. Get to know more about the domain and range of a relation.

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## Domain and Range of a Relation – Definition

Relation: If there are two non-empty sets A, B, then the relation r is defined as the subset of cross-product A x B. That subset is the result of the relation between the elements of both sets. R = {(a, b) : a ∈ A, b ∈ B}.

Domain: The set of elements of the first set belonging to the relation R is called the domain of the relation.

Domain(r) = {a : (a, b) ∈ R}

The domain of relation from A to B is a subset of A.

Range: The set of elements of the second set belonging to the relation R is called the range of the relation.

Range(r) = {b : (a, b) ∈ R}

The range of relation from A to B is a subset of B.

**Example:**

If two non empty sets are P = {1, 3, 7, 6, 5}, Q = {2, 4, 8, 9, 12}

Then R is the relation “less than” from set P to set Q

The relation R = {(1, 2), (1, 4), (3, 4), (7, 9), (5, 12), (6, 8)}

Therefore, Domain(R) = {1, 3, 5, 6, 7}, Range(R) = {2, 4, 8, 9, 12}

The arrow diagram to represent the domain and range of a relation is

### Domain and Range Examples and Answers

**Question 1:**

If A = {3, 5, 7, 6} and B = {21, 30, 18, 35, 10} and R be the relation ‘is factor of’ from A to B.

(i) Write R in the roster form and find the domain and range of a relation R.

(ii) Draw an arrow diagram to represent the relation.

**Solution:**

Given that,

A = {3, 5, 7, 6} and B = {21, 30, 18, 35, 10}

(i) R has elements of a, b and a is a factor of B. Where a is an element of A and b is an element of B.

Therefore, relation R in roster form = {(3, 21), (3, 18), (3, 30), (5, 30), (5, 35), (5, 10), (7, 21), (7, 35), (6, 18), (6, 30)}

The domain of the relation R is the set of all first components of R.

Range of relation R is the set of all second components of R.

So, Domain(R) = {3, 5, 6, 7}

Range(R) = {10, 18, 21, 30, 35}

(ii) The arrow diagram to represent R is

**Question 2:**

Find the domain and range of the relation R defined by R = {(x + 1, 2x + 3) : x ∈ {1, 2, 3, 4, 5}}

**Solution:**

Since x = {1, 2, 3, 4, 5}

Therefore,

x = 1 ⇒ x + 1 = 1 + 1 = 2 and 2x + 3 = 2(1) + 3 = 2 + 3 = 5

x = 2 ⇒ x + 1 = 2 + 1 = 3 and 2x + 3 = 2(2) + 3 = 4 + 3 = 7

x = 3 ⇒ x + 1 = 3 + 1 = 4 and 2x + 3 = 2(3) + 3 = 6 + 3 = 9

x = 4 ⇒ x + 1 = 4 + 1 = 5 and 2x + 3 = 2(4) + 3 = 8 + 3 = 11

x = 5 ⇒ x + 1 = 5 + 1 = 6 and 2x + 3 = 2(5) + 3 = 10 + 3 = 13

Hence relation R = {(2, 5), (3, 7), (4, 9), (5, 11), (6, 13)}

Domain of R = {a : (a, b) ∈R} = Set of first components of all ordered pair belonging to R.

Domain(R) = {2, 3, 4, 5, 6}

Range of R = {b : (a, b) ∈ R} = Set of second components of all ordered pairs belonging to R.

Range(R) = {5, 7, 9, 11, 13}

**Question 3:**

The arrow diagram shows the relation (R) from set A to set B. Write this relation in the roster form. Also, find the domain and range of the relation.

**Solution:**

R is the relation from A to B.

From the arrow diagram, we can write the relation R = {(13, q), (14, p), (16, r), (14, t), (18, q)}

The relation set R in the roster form = {(13, q), (14, p), (16, r), (14, t), (18, q)}

Domain of R = set of all first components of the relation = {13, 14, 16, 18}

Range of R = set of all second components of the relation = {p, q, r, t}

**Question 4:**

If X = {4, 5, 6, 8, 9, 10} define the relation R from X to X by R = {(x, y) : y = x + 1}

(i) Draw this relation using an arrow diagram.

(ii) Write domain and range of R.

**Solution:**

Given set X = {4, 5, 6, 8, 9, 10}

Relation R = {(x, y) : y = x + 1}

If x = 4, then y = x + 1 = 4 + 1 = 5

If x = 5, then y = x + 1 = 5 + 1 = 6

If x = 6, then y = x + 1 = 6 + 1 = 7

If x = 8, then y = x + 1 = 8 + 1 = 9

If x = 9, then y = x + 1 = 9 + 1 = 10

If x = 10, then y = x + 1 = 10 + 1 = 11

Relation R = {4, 5), (5, 6), (8, 9), (9, 10)}

(i) The arrow diagram of relation is

(ii) Domain of the relation = set of first components of all ordered pair belonging to R.

Therefore, domain of R = {4, 5, 8, 9}

Range of the relation = set of second components of all ordered pair belonging to R.

Therefore, range of R = {5, 6, 9, 10}.

### FAQs on Domain and Range of a Relation

**1. How do you state that domain and range of a relation?**

The domain of a relation is nothing but the set of all first components of R and the range of a relation is the set of all second components of R.

**2. What is a rule for relation?**

Relation a subset of the cartesian product of two sets. And it is a set of ordered pairs which has a relationship between the first set element with the second set element.

**3. Write an example for the domain and range?**

If relation R = {(1, 4), (2, 5), (3, 6), (4, 7)}, then the domain of relation R is {1, 2, 3, 4} and range of relation R is {4, 5, 6, 7}.