Question

Two finite sets have m and n elements. If the total number of subsets of first set is 56 more than the total number of subsets of second set, then find the values of m and n.

Answer 1

Let A has m elements

Let B has n elements

Total number of students of A$=2^m$

Total number of students of B$=2^n$

It is given $\Rightarrow 2^m-2^n=56$

$2^n(2^{m-n}-1)=56$

$\Rightarrow 2^n=$even and $2^{m-n}-1=0$ odd

Now,

$56=8\times 7=2^3\times 2^7$

$\Rightarrow 2^n(2^{m-n}-1)=2^3\times 7$

$\Rightarrow n=3$

Now, $8(2^{m-3}-1)=8\times 7$

$\Rightarrow 2^{m-3}-1=7$

$\Rightarrow 2^{m-3}=8=2^3$

$\Rightarrow m-3=3$

$\Rightarrow m=6$.