Question

Two finite setshave $m$ and $n$ elements. The number of subsets of the first set is 112 more than that of the second set. The values of $m$ and $n$ are, respectively,

• A. 7,4
• B. 4,7
• C. 7,7
• D. 4,4

Answer 1

The correct option is A 7,4

Given: Two finite sets having m,n
elements
Also, the no. of subsets of the
first set is 112 more than that
of the second set.

Let A is a set having $m$ elements and $B$ is a set having $n$ elements

$\therefore \text{Number of subsets of}A=2^m$

$\text{And number of subsets of}B=2^n$

$\therefore 2^m-2^n=112[\text{Given]}$

$\Rightarrow 2^n(2^{m-n}-1)=2^4.7$
Comparing both sides, we get

$\Rightarrow 2^n=2^4$ $2^{m-n}-1=7$

$\Rightarrow n=4$ $\Rightarrow 2^{m-n}-1=8-1$

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

$\Rightarrow n=4\text{and}m-n=3$
$\Rightarrow m=7$

$\therefore$ The value of $m,nis7,4$ respectively.

Hence, option (B) is correct.