Question

$A$ and $B$ are two finite sets, such that $A$ has $p$ elements and $B$ has $q$ elements. The number of elements in the power set of $A$ is $48$ more than number of elements in the power set of $B.$ Then select the correct statement about $p\&q.$

• A. $p=4$
• B. $p=6$
• C. $q=6$
• D. $q=4$

Answer 1

Answer: (B), (D)

$q=4$

Given $n\left(A\right)=p;n\left(B\right)=q$
Now, Cardianality of power set of any set $X$ having $x$ elements is given as:
$n\left(X\right)=x$
$\Rightarrow n\left(P\right(A\left)\right)=2^p;n\left(P\right(B\left)\right)=2^q$
It is given, that $2^p=48+2^q$
$\Rightarrow 2^p-2^q=48$
$\Rightarrow 2^q(\frac{2^p}{2^q}-1)=48$
$\Rightarrow 2^q(2^{p-q}-1)=48$
$\Rightarrow 2^q(2^{p-q}-1)=2^4(2^2-1)$
Comparing, we get:
$q=4;p=q+2=6$