If A = {1, 2, 3, ...., 14} and R is a relation defined on A such that R = {(x, y) : 3x - y = 0 where x, y ∈ A} then find the range of R ?

Concept:

Range of a Relation: Let R be a relation from set A to set B. Then, the set of all second components of the ordered pair belonging to relation R forms the range of the relation R.

i.e Range (R) = {b: (a, b) ∈ R}.

Calculation:

Given: A = {1, 2, 3, ...., 14} and R is a relation defined on A such that R = {(x, y) : 3x - y = 0 where x, y ∈ A}

When x = 1 ∈ A then y = 3x = 3 ∈ A ⇒ (1, 3) ∈ R

When x = 2 ∈ A then y = 3x = 6 ∈ A ⇒ (2, 6) ∈ R

When x = 3 ∈ A then y = 3x = 9 ∈ A ⇒ (3, 9) ∈ R

When x = 4 ∈ A then y = 3x = 12 ∈ A ⇒ (4, 12) ∈ R

So, R = {(1, 3), (2, 6), (3, 9), (4, 12)}

As we know that Range (R) = {b: (a, b) ∈ R}

⇒ Range(R) = {3, 6, 9, 12}

Hence, option 1 is the correct answer.