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 : 2x + 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}
= {x : x ∊ N and x > 9/2}
∴ A′ = U – A = {x : x ∈ N and x ≤ 9/2}