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Q.
Let $A$ and $B$ be two sets. The set $A$ has $2016$ more subsets than $B.$ If $A\cap B$ has $3$ members, then the number of members in $A\cup B$ is
NTA AbhyasNTA Abhyas 2020
Solution:
If $A$ has $m$ members and $B$ has $n$ members, then
$2^{m}-2^{n}=2016=2^{5}.63$
$\Rightarrow 2^{n}\left(2^{m - n} - 1\right)=2^{5}\left(64 - 1\right)=2^{5}\left(2^{6} - 1\right)$
$\Rightarrow n=5$ and $m-n=6\Rightarrow m=11$
The number of members in $A\cup B$
$=11+5-3=13$