Question

If $n\left(A\right)=3,n\left(B\right)=5$and $n(A\cap B)=2$ then $n\left[\right(A\times B)\cap (B\times A\left)\right]=$

• A. $5$
• B. $3$
• C. $4$
• D. $6$

Answer 1

The correct option is C

$4$

Explanation of the correct option

Given: $n\left(A\right)=3,n\left(B\right)=5,$$n(A\cap B)=2$

We know that,

$$\begin{gathered} (A\times B)\cap (B\times A)=\left(A\cap B\right)\times \left(B\cap A\right) \\ \Rightarrow (A\times B)\cap (B\times A)={\left(A\cap B\right)}^2\left[\because \left(A\cap B\right)=\left(B\cap A\right)\right] \\ \Rightarrow n\left[(A\times B)\cap (B\times A)\right]={\left[n\left(A\cap B\right)\right]}^2 \\ \Rightarrow n\left[(A\times B)\cap (B\times A)\right]=2^2 \\ \Rightarrow n\left[(A\times B)\cap (B\times A)\right]=4 \end{gathered}$$

Hence, option(C) i.e. $4$ is the correct option.