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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 the set A ?
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The correct answer is:
6
Let sets $A$ and $B$ have $m$ and $n$ elements respectively. The set made by subsets of a finite sets $A$ and $B_{\underline{1}}$ s known as power set i.e. $\mathrm{P}(\mathrm{A})$ and $\mathrm{P}(\mathrm{B})$ We know, if set $\mathrm{A}$ has $n$ elements then $\mathrm{P}(\mathrm{A})$ has $2^{n}$ elements. Thus, The total no. of subsets of a finite set $\mathrm{A}=2^{m}$
and set $B=2^{n}$
So, According to the question $2^{m}-2^{n}=56$
$\Rightarrow 2^{n}\left(2^{m-n}-1\right)=8 \times 7=2^{3} \times\left(2^{3}-1\right)$
On comparing the powers, both side $n=3$ and $m-n=3$
$\Rightarrow m=6$ and $n=3$
and set $B=2^{n}$
So, According to the question $2^{m}-2^{n}=56$
$\Rightarrow 2^{n}\left(2^{m-n}-1\right)=8 \times 7=2^{3} \times\left(2^{3}-1\right)$
On comparing the powers, both side $n=3$ and $m-n=3$
$\Rightarrow m=6$ and $n=3$
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