Question

Taking the set of natural numbers as the universal set, write down the complement 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 $\ge$ 7}

(xi) {x : x $\in$ N and 2x + 1 > 10}

Answer 1

Here U = {x : x $\in$ N}

(i) Let A = {x : x is an even natural number}

$A^{\prime }$ = U - A

= {x : x $\in$ 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}

$A^{\prime }$ = U - A

= {x : x $\in$ 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}

$\therefore A^{\prime }$ = U - A

= {x : x $\in$ N} - {x : x is a positive multiple of 3}

= {x : x $\in$ N, x is not a multiple of 3}

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

$A^{\prime }$ = U - A

= {x : x $\in$ N} - {x : x is a prime number}

= {x : x $\in$ 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}

$A^{\prime }$ = U - A

= {x : x $\in$ N} - {x : x is a natural number divisible by 3 and 5}

= {x : x $\in$ N} - {x : x is a natural number divisible by 15}

= {x : x $\in$ N, x is not divisible by 15}

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

$A^{\prime }$ = U - A

= {x : x $\in$ N} - {x : x is a perfect square}

= {x : x $\in$ N, x is not a perfect square}

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

$A^{\prime }$ = U - A

= {x : x $\in$ N} - {x : x is a perfect cube}

= {x : x $\in$ N, x is not a perfect cube}

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

$A^{\prime }$ = U - A

= {x : x $\in$ N} - {3} = {x : x $\in$ N, x $\ne$ 3}

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

$A^{\prime }$ = U - A = {x : x $\in$ N} - {2}

= {x : x $\in$ N, x $\ne$ 2}

(x) Let A = {x : x $\ge$ 7} = {7, 8, 9, 10, ..........}

$A^{\prime }$ = U - A

= {x : x $\in$ N} - {7, 8, 9, 10, ..........}

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

= {x : x $\in$ N and x < 7}

(xi) Let A = {x : x $\in$ N and 2x + 1 > 10}

= {5, 6, 7, 8, .........}

$A^{\prime }$ = U - A

= {x : x $\in$ N} - {5, 6, 7, 8,..........}

= {1, 2, 3, 4}