Question:
Set A has $m$ elements and set B has $n$ elements. If the total number of subsets of $A$ is 112 more than the total number of subsets of B, then the value of $m \cdot n$ is___________.
Solution:
$2^{m}=112+2^{n} \Rightarrow 2^{m}-2^{n}=112$
$\Rightarrow 2^{n}\left(2^{m-n}-1\right)=2^{4}\left(2^{3}-1\right)$
$\therefore m=7, n=4 \Rightarrow m n=28$
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