Question

Find the domain and range of the following relations

(i) R₁ = {(x, y) : x, y ∈ N, x + y = 10}

(ii) R₂ = {(x, y) : y = |x – 1|, x ∈ z and |x| ≤ 3}

Answer 1

(i) Given set N = {1, 2, 3,…}

A relation R₁ in N is defined as

R₁ = {(x, y) : x, y ∈ N, x + y = 10} ∀ x, y ∈ N

So, xR₁y ⇔ x + y = 10 ⇔ y = 10 – x

when x = 1, then y = 10 – 1 = 9 ∈ N then (1, 9) ∈ R₁

when x = 2, then y = 10 – 2 = 8 ∈ N then (2, 8) ∈ N

when x = 3, then y = 10 – 3 = 7 ∈ N then (3, 7) ∈ R₁

when x = 4, then y = 10 – 4 = 6 ∈ N then (4, 6) ∈ R₁

Similarly, (5, 5) ∈ R₁, (6, 4) ∈ R₁, (7, 3) ∈ R₁, (8, 2) ∈ R₁, (9, 1) ∈ R₁

R₁ = {(1, 9), (2, 8), (3, 7), (4, 6), (5, 5), (6, 4), (7, 3), (8, 2), (9, 1)}

Domain of R₁ ={1, 2, 3, 4, 5, 6, 7, 8, 9}

Range of R₁ = {9, 8, 7, 6, 5, 4, 3, 2, 1}

(ii) Given set Z = set of integers = {0, ±1, ±2, ±3,…}

A relation R in Z is defined as

R₂ = {(x, y), y = |x – 1|, x ∈ z and x ≤ 3}

or xRy ⇔ y = |x – 1|, |x| ≤ 3 ∀ y ∈ Z

Here |x| ≤ 3 ⇒ -3 ≤ x ≤ 3, x ∈ Z

Domain of R₂ = {-3, -2, -1, 0, 1, 2, 3}

Range of R₂ = {4, 3, 2, 1, 0}