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Question

# Taking the set of natural numbers as the universal set, write down the complement of the following sets : (i) {x : x is an even natural number} (ii) {x : x is an odd natural number} (iii) {x : x is a positive multiple of 3} (iv) {x : x is a prime number} (v) {x : x is a natural number divisible by 3 and 5} (vi) {x : x is a perfect square} (vii) {x : x is a perfect cube} (viii) {x : x + 5 = 8} (ix) {x : 2x + 5 = 9} (x) {x : x ≥ 7} (xi) {x : x ∈ N and 2x + 1 > 10}

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Solution

## Here U = {x : x ∈ N} (i) Let A = {x : x is an even natural number} A′ = U - A = {x : x ∈ N} - {x : x is an even natural number} = {x : x is an odd natural number} (ii) Let A = {x : x is an odd natural number} A′ = U - A = {x : x ∈ N} - {x : x is an odd natural number} = {x : x is an even natural number} (iii) Let A = {x : x is a positive multiple of 3} ∴ A′ = U - A = {x : x ∈ N} - {x : x is a positive multiple of 3} = {x : x ∈ N, x is not a multiple of 3} (iv) Let A = {x : x is a prime number} A′ = U - A = {x : x ∈ N} - {x : x is a prime number} = {x : x ∈ N, x is not a prime number} or {x : x is positive composite number and x = 1} (v) Let A = {x : x is a natural number divisible by 3 and 5} A′ = U - A = {x : x ∈ N} - {x : x is a natural number divisible by 3 and 5} = {x : x ∈ N} - {x : x is a natural number divisible by 15} = {x : x ∈ N, x is not divisible by 15} (vi) Let A = {x : x is a perfect square} A′ = U - A = {x : x ∈ N} - {x : x is a perfect square} = {x : x ∈ N, x is not a perfect square} (vii) Let A = {x : x is a perfect cube} A′ = U - A = {x : x ∈ N} - {x : x is a perfect cube} = {x : x ∈ N, x is not a perfect cube} (viii) Let A = {x : x + 5 = 8} = {3} A′ = U - A = {x : x ∈ N} - {3} = {x : x ∈ N, x ≠ 3} (ix) Let A = {x : 2 x + 5 = 9} = {2} A′ = U - A = {x : x ∈ N} - {2} = {x : x ∈ N, x ≠ 2} (x) Let A = {x : x ≥ 7} = {7, 8, 9, 10, ..........} A′ = U - A = {x : x ∈ N} - {7, 8, 9, 10, ..........} = {1, 2, 3, 4, 5, 6} = {x : x ∈ N and x < 7} (xi) Let A = {x : x ∈ N and 2x + 1 > 10} = {5, 6, 7, 8, .........} A′ = U - A = {x : x ∈ N} - {5, 6, 7, 8,..........} = {1, 2, 3, 4}

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