Question

If $X$ and $Y$ are two sets such that $(X\cup Y)$ has $60$ elements, $X$ has $38$ elements and $Y$ has $42$ elements, how many elements do $(X\cap Y)$ have?

• A. $11$
• B. $20$
• C. $13$
• D. None of these

Answer 1

The correct option is B

$20$

Explanation for the correct option.

We know that the relation between the number of elements in a set is as follows;

$$\begin{gathered} \begin{array}{rcl}n\left(X\cup Y\right)& =& n\left(X\right)+n\left(Y\right)-n\left(X\cap Y\right)\\ \Rightarrow 60& =& 38+42-n\left(X\cap Y\right)\\ & \Rightarrow & n\left(X\cap Y\right)=80-60\end{array} \\ \Rightarrow n(X\cap Y)=20 \end{gathered}$$

Hence, option B is correct.