Question

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

Answer 1

Given, universal set is the set of natural numbers.

(i)

The given set is { x:x is an even natural number } .

Therefore, complement of this set will be the set of all odd natural numbers.

Thus, Complement = { x:x is an odd natural number } .

(ii)

The given set is { x:x is an odd natural number } .

Therefore, complement of this set will be the set of all even natural numbers.

Thus, Complement = { x:x is an even natural number } .

(iii)

The given set is { x:x is a positive multiple of 3 } .

Therefore, complement of this set will have values of x not equal to a positive multiple of 3.

Thus, Complement = { x:x is not a positive multiple of 3 } .

(iv)

The given set is { x:x is a prime number } .

Therefore, complement of this set will have values of x which is not a prime number.

Thus, Complement = { x:x is not a prime number } .

(v)

The given set is { x:x is a natural number divisible by 3 and 5 } .

Therefore, complement of this set will have values of x which are not divisible by 3 and 5.

Thus, Complement = { x:x is a natural number not divisible by 3 and 5 } .

(vi)

The given set is { x:x is a perfect square } .

Therefore, complement of this set will have values of x which are not a perfect square.

Thus, Complement = { x:x is not a perfect square } .

(vii)

The given set is { x:x is a perfect cube } .

Therefore, complement of this set will have values of x which are not a perfect cube.

Thus, Complement = { x:x is not a perfect cube } .

(viii)

The given set is { x:x+5=8 } .

Solving for x, x+5=8 x=3

Therefore, the complement of this set will have values of x not equal to 3.

Thus, Complement = { x:x is a natural number not equal to 3 } .

(ix)

The given set is { x:2x+5=9 } .

Solving for x, 2x+5=9 2x=4 x=2

Therefore, complement of this set will have values of x not equal to 2.v

Thus, Complement = { x:x is a natural number not equal to 2 } .

(x)

The given set is { x:x≥7 } .

Therefore, complement of this set will have values of x less than 7, i.e. 1,2,3,4,5,6 .

Thus, Complement = { x:x=1,2,3,4,5,6 } .

(xi)

The given set is { x:x∈N and 2x+1>10 } .

Solving for x, 2x+1>10 2x>9 x> 9 2

Therefore, complement of this set will have values of x less than or equal to 4.5, i.e. 1,2,3,4 .

Thus, Complement = { x:x∈N and x≤4 } .