Question

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?

Answer 1

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