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If $X$ and $Y$ are two sets such that $X \cup Y$ has 18 elements, $X$ has 8 elements and $Y$ has 15 elements; how many elements does $X \cap Y$ have?
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Here $n(X)=8$ and $n(Y)=15$ and $n(X \cup Y)=18$ $\therefore \quad n(X \cup Y)=n(X)+n(Y)-n(X \cap Y)$
$\Rightarrow 18=8+15-n(X \cap Y) \Rightarrow n(X \cap Y)=23-18=5$
$\Rightarrow 18=8+15-n(X \cap Y) \Rightarrow n(X \cap Y)=23-18=5$
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