If A = {3, 9, 27, 81}, B = {1, 2, 3} and R is a relation from A to B such that R = {(x, y) : x ∈ A, y ∈ B and y = log₃ x} then find the number of elements in the domain of R ?

Concept:

Domain of a Relation: 

Let R be a relation from set A to set B. Then, the set of all first components of the ordered pair belonging to relation R forms the domain of the relation R

i.e Domain (R) = {a : (a, b) ∈ R}.

Calculation:

Given: A = {3, 9, 27, 81}, B = {1, 2, 3} and R is a relation from A to B such that R = {(x, y) : x ∈ A, y ∈ B and y = log3 x}

As we know that, logx x = 1 for x > 0

∵ R = {(x, y) : x ∈ A, y ∈ B and y = log3 x}

When x = 3 ∈ A then y = log3 3 = 1 ∈ B ⇒ (3, 1) ∈ R

When x = 9 ∈ A then y = log3 9 = 2 ∈ B ⇒ (9, 2) ∈ R

When x = 27 ∈ A then y = log3 27 = 3 ∈ B ⇒ (27, 3) ∈ R

When x = 81 ∈ A then y = log3 81 = 4 ∉ B ⇒ (81, 4) ∉ R

So, the given relation R can be re-written in roster form as: R = {(3, 1), (9, 2), (27, 3)} 

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

⇒ Domain (R) = {3, 9, 27}

So, the number of elements in the domain of given relation R is 3.