If A = {2, 4, 8, 16}, B = {1, 2, 3, 4, 5, 6} and R is a relation from A to B such that R = {(x, y) : x ∈ A, y ∈ B and y = log₂ x} 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 = {2, 4, 8, 16}, B = {1, 2, 3, 4, 5, 6} and R is a relation from A to B such that R = {(x, y) : x ∈ A, y ∈ B and y = log2 x}

As we know that, log_x x = 1 for x > 0

∵ R = {(x, y) : x ∈ A, y ∈ B and y = log2 x}

When x = 2 ∈ A then y = log₂ 2 = 1 ∈ B ⇒ (2, 1) ∈ R

When x = 4 ∈ A then y = log₂ 4 = 2 ∈ B ⇒ (4, 2) ∈ R

When x = 8 ∈ A then y = log₂ 8 = 3 ∈ B ⇒ (8, 3) ∈ R

When x = 16 ∈ A then y = log₂ 16 = 4 ∈ B ⇒ (16, 4) ∈ R

So, the given relation R can be re-written in roaster form as: R = {(2, 1), (4, 2), (8, 3), (16, 4)}

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

⇒ Range (R) = {1, 2, 3, 4}

Hence, the correct option is 1.