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) $\{x:x\ge 7\}$

(xi) $\{x:xϵNand2x+1>10\}$

Answer 1

U= N: Set of natural numbers
(i) $\{x:x\text{is an even natural number}{\}}^{\prime }=\{x:x\text{is an odd natural number}\}$
(ii) $\{x:x\text{is an odd natural number}{\}}^{\prime }=\{x:x\text{is an even natural number}\}$
(iii) $\{x:x\text{is a positive multiple of 3}{\}}^{\prime }=\{x:x\in N\text{and x is not a multiple of 3}\}$
(iv) $\{x:x\text{is a prime number}{\}}^{\prime }=\{x:x\text{is a positive composite number and x=1}\}$
(v) $\{x:x\text{is a natural number divisible by 3 and 5}{\}}^{\prime }$

$=\{x:x\text{is a natural number that is not divisible by 3 or 5}\}$
(vi) $\{x:x\text{is a perfect square}{\}}^{\prime }=\{x:x\in N\text{and x is not a perfect square}\}$
(vii) $\{x:x\text{is a perfect cube}{\}}^{\prime }=\{x:x\in N\text{and x is not a perfect cube}\}$
(viii) ${\{x:x+5=8\}}^{\prime }=\{x:x\in N\text{and}x\ne 3\}$
(ix) ${\{x:2x+5=9\}}^{\prime }=\{x:x\in N\text{and}x\ne 2\}$
(x) ${\{x:x\ge 7\}}^{\prime }=\{x:x\in N\text{and}x<7\}$
(xi) ${\{x:xϵN\text{and}2x+1>10\}}^{\prime }=\{x:x\in Nandx\le \frac{9}{2}\}$