Taking the set of natural numbers as the universal set, write down the complements 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}

For all parts, given that
(i) Let A = {x : x is an even natural number}

We want to find complement of A , which is given by U - A

⇒ A’ = U - A

⇒ A’ = {x:x ϵ N} - {x : x is an even natural number}

⇒ A’ = {x : x is an odd natural number}

(ii) Let A = {x : x is an odd natural number}

⇒ A’ = U - A

⇒ A’ = {x:x ϵ N} - {x : x is an odd natural number}

⇒ A’ = {x : x is an even natural number}

(iii) Let A = {x : x is a positive multiple of 3}

⇒ A’ = U - A

⇒ A’ = {x:x ϵ N} - {x : x is a positive multiple of 3}

⇒ A’ = {x : x is not a positive multiple of 3}

(iv) Let A = {x : x is a prime number}

⇒ A’ = U - A

⇒ A’ = {x: x ϵ N} - {x : x is a prime number}

⇒ A’ = {x : x is not a prime number}

(v) Let A = {x : x is a natural number divisible by 3 and 5}

∴ A = {x : x is a natural number divisible by 15}

⇒ A’ = U - A

⇒ A’ = {x: x ϵ N} - {x : x is a natural number divisible by 15}

⇒ A’ = {x : x is a natural number not divisible by 15}

(vi) Let A = {x : x is a perfect square}

⇒ A’ = U - A

⇒ A’ = {x: x ϵ N} - {x : x is a perfect square}

⇒ A’ = {x : x is not a perfect square}

(vii) Let A = {x : x is a perfect cube}

⇒ A’ = U - A

⇒ A’ = {x: x ϵ N} - {x : x is a perfect cube}

⇒ A’ = {x : x is not a perfect cube}

(viii) Let A = {x : x + 5 = 8}

∴ A = {x : x = 3}

⇒ A’ = U - A

⇒ A’ = {x: x ϵ N} - {x : x = 3}

⇒ A’ = {x : x ϵ N and x ≠ 3}

(ix) Let A = {x : 2x + 5 = 9}

∴ A = {x : x = 2}

⇒ A’ = U - A

⇒ A’ = {x: x ϵ N} - {x : x = 2}

⇒ A’ = {x : x ϵ N and x ≠ 2}

(x) Let A = {x : x ≥ 7}

⇒ A’ = U - A

⇒ A’ = {x: x ϵ N} - {x : x ≥ 7}

⇒ A’ = {x : x < 7}

(xi) Let A = {x: x ϵ N} - {x : 2x + 1 > 10}

∴ A = {x: x ϵ N and x > 9/2}

⇒ A’ = U - A

A’ = {x:x ϵ N} - {x: x ϵ N and x > 9/2}

⇒ A’ = {x: x ϵ N and x < 9/2}

∴ A’ = {1, 2, 3, 4}