If X and Y are two sets such that X ∪Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X ∩Y have?

Question:

If X and Y are two sets such that X Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X Y have?

Solution:

It is given that:

n(X ∩ Y) = ?

We know that:

$n(\mathrm{X} \cup \mathrm{Y})=n(\mathrm{X})+n(\mathrm{Y})-n(\mathrm{X} \cap \mathrm{Y})$

$\therefore 18=8+15-n(\mathrm{X} \cap \mathrm{Y})$

$\Rightarrow n(\mathrm{X} \cap \mathrm{Y})=23-18=5$

$\therefore n(\mathrm{X} \cap \mathrm{Y})=5$