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

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 perfect cube}
(viii) {x: x + 5 = 8}
(ix) {x: 2x + 5 = 9}
(x) {x: x ≥ 7}
(xi) {x: x ∈ N and 2x + 1 > 10}

Answer

Universal set ⇒ Set of natural numbers ⇒ {1, 2, 3, 4, 5...}
Set of even natural numbers ⇒ {2, 4, 6, 8 ...}
Set of odd natural number ⇒ {1, 3, 5...}
Set of prime number ⇒ {2, 3, 5, 7...}

(i) Complements of {x: x is an even natural number}
⇒ {1, 2, 3, 4, 5...} - {2, 4, 6, 8 ...}
⇒ {1, 3, 5...}
⇒ {x: x is an odd natural number}
(ii) Complements of {x: x is an odd natural number}
⇒ {1, 2, 3, 4, 5...} - {1, 3, 5...}
⇒ {2, 4, 6, 8 ...}
⇒ {x: x is an even natural number}
(iii) Complements of {x: x is a positive multiple of 3} 
⇒ {x: x ∈ N and x is not a multiple of 3}

(iv) Complements of {x: x is a prime number} 
⇒{x: x is a positive composite number and x = 1}
(v) Complements of {x: x is a natural number divisible by 3 and 5} 
⇒ {x: x is a natural number that is not divisible by 3 or 5}
(vi) Complements of {x: x is a perfect square} 
⇒ {x: x ∈ N and x is not a perfect square}
(vii) Complements of {x: x is a perfect cube} 
⇒ {x: x ∈ N and x is not a perfect cube}
(viii){x: x + 5 = 8}
Solving the equation x + 5 ⇒ 8 we get x = 3.
Complement of this set will not have x = 3, Hence complement set can be {x: x ∈ N and x ≠ 3}
(ix){x: 2x + 5 = 9}
Solving the equation 2x + 5 = 9 we get x = 2.
Complement of this set will not have x = 2,
Hence complement set can be {x: x ∈ N and x ≠ 2}
(x) Complements of {x: x ≥ 7} 
={x: x ∈ N and x < 7}
(xi) Complements of {x: x ∈ N and 2x + 1 > 10}
Solve the equation
2x + 1 > 10
2x > 9
X > 9/2
All values of x more than 9/2 cannot be in set of complement.
Hence, complements of {x: x ∈ N and 2x + 1 > 10} ⇒ {x: x ∈ N and x ≤ 9/2}