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?
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?
It is given that:
$n(X \cup Y)=18, n(X)=8, n(Y)=15$
$n(X \cap Y)=?$
We know that:
$n(X \cup Y)=n(X)+n(Y)-n(X \cap Y)$
$\therefore 18=8+15-n(X \cap Y)$
$\Rightarrow n(X \cap Y)=23-18=5$
$\therefore n(X \cap Y)=5$
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