Finding The Power Set

Number of subset of a set with $n$ element is $2^{n}$ . We know that

$192 = 2^{6}*3 = 2^{6}(2^{2}-1) = 2^{8} - 2^{6}$

therefore, we could conclude that $A$ have $8$ elements while $B$ have $6$ elements.

Assume that the set $A$ contains $a$ elements and the set $B$ contains $b$ elements. Clearly $b \leq a-1$ . Thus we have, $\frac{1}{2}2^a \leq 2^a-2^b=192 \leq 2^a$ Since $a$ is an integer, the above inequality has the unique solution $a=8$ .