Question

The total number of subsets of a finite set $A$ has $56$ more elements than the total number of subsets of another finite set $B$. What is the number of elements in set $A$ ?

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is B

$6$

Explanation of the correct option.

Let the number of elements in set $A$ be $p$ and set $B$ be $q$.

The power set of $A$ is $P\left(A\right)$ and $B$ is $P\left(B\right)$.

$P\left(A\right)$ has $2p$ elements and $P\left(B\right)$ has $2q$ elements.

Thus, $2p-2q=56$

$$\begin{gathered} \Rightarrow 2^q[2^{p-q}–1]=8\times 7 \\ \Rightarrow 2^q\left[2^{p-q}–1\right]=2^3\left[2^3-1\right] \end{gathered}$$

Comparing L.H.S. and R.H.S.

$q=3,p–q=3$

$\therefore$$p-3=3$

$\Rightarrow$ $p=6$

Therefore number of elements in set $A$ is $6$.

Hence $\mathrm{Option}\left(\mathrm{B}\right)$ is the correct option.