Question

Taking the set of natural numbers as the universal set, write down the complements of the following set:
$\left(i\right)\{x:x$ is an even natural number $\}$
$\left(ii\right)\{x:x$ is an odd narutal number$\}$
$\left(iii\right)\{x:x$ is a positive multiple of $3\}$
$\left(iv\right)\{x:x\text{is a prime number}\}$
$\left(v\right)\{x:x\text{is a natural number divisible by 3 and 5}\}$
$\left(vi\right)\{x:x\text{is a perfect square}\}$
$\left(vii\right)\{x:x\text{is a perfect cube}\}$
$\left(viii\right)\{x:x+5=8\}$
$\left(ix\right)\{x:2x+5=9\}$
$\left(x\right)\{x:x\ge 7\}$
$\left(xi\right)\{x:x\in N\text{and}2x+1>10\}$

Answer 1

$\left(i\right)$ Given: Universal set $\left(U\right)=\{1,2,3,4,5,6,...\}$
Let $A=\{x:x$ is an even natural number $\}=\{2,4,6,8,...\}$
$\therefore$ Compliment of given set.
$A^{\prime }=U-A$
$=\{1,2,3,4...\}-\{2,4,6,8,...\}$
$=\{1,3,5,7,9\}$
$=\{x:x$ is an odd narutal number$\}$

$\left(ii\right)$ Given: Universal set $\left(U\right)=\{1,2,3,4,5,6,...\}$
Let $B=\{x:x$ is an odd narutal number$\}=\{1,3,5,7,9,...\}$
$\therefore$ Compliment of given set,
$B^{\prime }=U-B$
$=\{1,2,3,4,5,6,7,8,9,10,...\}-\{1,3,5,7,9,...\}$
$=\{2,4,6,8,10,...\}$
$=\{x:x$ is an even natural number $\}$

$\left(iii\right)$ Given: Universal set $\left(U\right)=\{1,2,3,4,5,6,...\}$
Let $C=\{x:x$ is a positive multiple of $3\}=\{3,6,9,12,....\}$
$\therefore$ Complimnet of given set,
$C^{\prime }=U-C$
$=\{1,2,3,4,5,6,7,8,9,...\}-\{3,6,9,12,15,...\}$
$=\{x:x$ is not a multiple of $3$ and $x\in N\}$

$\left(iv\right)$ Given: Universal set $\left(U\right)=\{1,2,3,4,5,6,...\}$
Let $D=\{x:x$ is a prime number $\}=\{2,3,5,7,11,13,...\}$
$\therefore$ Compliment of given set,
$D^{\prime }=U-D$
$=\{1,2,3,4,5,6,7,8,9,...\}-\{2,3,5,7,11,13,...\}$
$=\{1,4,6,8,9,...\}$
$=\{x:x$ is a composite number or $x=1$ and $x\in N\}$

$\left(v\right)$ Given: Universal set $\left(U\right)=\{1,2,3,4,5,6,...\}$
Let $E=\{x:x$ is a natural number that is divisible by $3$ and $5\}$
$\therefore$ Compliment of given set,
$E^{\prime }=\{x:x$ is a natural number that is not divisible by $3$ and $5\}$

$\left(vi\right)$ Given universal set $\left(U\right)=\{1,2,3,4,5,6,...\}$
Let $F=\{x:x$ is a perfect square$\}$
$\therefore$ Compliment of given set,
$F^{\prime }=\{x:x$ is not a perfect square and $x\in N\}$

$\left(vii\right)$ Given: Universal set $\left(U\right)=\{1,2,3,4,5,6,...\}$
Let $G=\{x:x\}$ is a perfect cube $\}=\{1,8,27,64,...\}$
$\therefore$ Compliment of given set
$G^{\prime }=\{x:x$ is not a perfect cube $x\in N\}$

$\left(viii\right)$ Given: $U=\{1,2,3,4,5,6,...\}$
Let $H=\{x:x+5=8\}$
$\Rightarrow x+5=8$
$\Rightarrow x=8-5$
$\Rightarrow x=3$
$\Rightarrow H=\left\{3\right\}$
$\therefore$ Compliment of given set,
$H^{\prime }=\{x:x\in N\text{and}x\ne 3.\}$

$\left(ix\right)$ Given: $U=\{1,2,3,4,5,...\}$
Let $I=\{x:2x+5=9\}$
$\Rightarrow 2x+5=9$
$\Rightarrow 2x=9-5$
$\Rightarrow x=\frac{4}{2}$
$\Rightarrow x=2$
$\Rightarrow I=\left\{2\right\}$
$\therefore$ Compliment of given set
$I^{\prime }=\{x:x\in N\text{and}x\ne 2\}$

$\left(x\right)$ Given $U=\{1,2,3,4,5,...\}$
Let $J=\{x:x\ge 7\}=\{7,8,9,10,...\}$
$\therefore$ Compliment of given set
$J^{\prime }=\{x:x<7\text{and}x\in N\}$

$\left(xi\right)$ Given: $U=\{1,2,3,4,5,6...\}$
Let $K=\{x:x\in N\text{and}2x+1>10\}$
$\Rightarrow 2x+1>10$
$\Rightarrow 2x>10-1$
$\Rightarrow x>\frac{9}{2}$
$\Rightarrow x>4.5$
$\Rightarrow K=\{5,6,7,8,...\}$
$\therefore$ Compliment of given set
$K^{\prime }=\{1,2,3,4\}=\{x:x<4.5\text{and}x\in N\}$