Question

A relation on the set $A=[x:\mid x\mid <3,xϵZ]$, Where $Z$, the set of integers, is defined by $R=\left[\right(x,y):y=\mid x\mid ,x\ne -1]$. Then, the number of elements in the power set of $R$ is

• A. $32$
• B. $16$
• C. $8$
• D. $64$

Answer 1

The correct option is B $16$

First consider $R=\left[\right(x,y):y=\mid x\mid ,x\ne -1]$
$A=\{-2,-1,0,1,2\}$ (since $|x|<3$ and $x\in Z$)
$\Rightarrow R=\left\{\right(0,0),(1,1),(2,2),(-2,2\left)\right\}$ (since $x\ne -1$)
There are $4$ elements in $R$.
Power set contains all subsets of a given set.
Hence number of elements in power set of $R$
${=}^4{C}_0{+}^4{C}_1{+}^4{C}_2{+}^4{C}_3{+}^4{C}_4$
$=1+4+6+4+1=16$.
Alternatively, using the result, No. of elements in power set$=2^n$,
where '$n$' is the total number of elements in a set,
we get $2^4=16$.