Relations and Functions

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 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}

Answer

(i) A = {x : x is na even atural number} =

∴ A ' = U - A =

= {x : x is an odd natural number}

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

A' = U - A = - {x : x is an odd number}

= {x : x is an even natural number} =

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

A' = U - A = {1, 2, 3, 4, 5, 6, .......} - {3, 6, 9, 12, .........}

= {1, 2, 4, 5, ...........} = , x is not a multiple of 3}

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

A' = U - A = - x is a prime number

= , x is not a prime number}

(v) A = , x is divisible by 3 and 5}

=, x is divisible by 15.

A' = U - A = , x is divisible by 15}

= , x is not divisible by 15}

= , x is not divisible by 3 and 5}

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

A' = U - A = - (x : x is a perfect square)

= , x is not a perfect square}

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

A' = U - A = - {x : x is a perfect cube}

= , x is not a perfect cube}

(viii) A = {x : x +5 = 8} = {x : x = 3} = {3}

∴ A' = U - A = {1, 2, 3, 4, 5, 6, .........} - {3} = {1, 2, 3, 4, 5, 6, .........}

∴ A' =

(ix) A = { x : 2x + 5 = 9} = {x : 2x = 4} = {x : x = 2} = {2}

∴ A' = U - A = {1, 2, 3, 4, 5, 6, ..........} - {3} = {1, 2, 4, 5, 6, ............}

(x) A =

A' = U - A =

= {1, 2, 3, 4, 5, 6}

(xi)

∴ A' = U - A = {1, 2, 3, 4, 5, 6, 7, 8, .........} - {5, 6, 7, 8, ..........}

= {1, 2, 3, 4}