Question

If $n\left(A\right)$denotes the number of elements in the set $A$ and if $n\left(A\right)=4$, $n\left(B\right)=5$ and $n(A\cap B)=3$, then$n\left[\right(A\times B)\cap (B\times A\left)\right]$ is equal to

• A. $8$
• B. $9$
• C. $10$
• D. $11$

Answer 1

The correct option is B

$9$

Explanation for the correct option

Given that $n\left(A\right)=4$, $n\left(B\right)=5$ and $n(A\cap B)=3$

For $(A\times B)\cap (B\times A)$ we have to do the mapping of$A\times B$ or $B\times A$between common elements.

$\Rightarrow$$n\left((A\times B)\cap (B\times A)\right)=n\left(\left(A\cap B\right)\times \left(B\cap B\right)\times \left(B\cap A\right)\times \left(A\cap A\right)\right)$

We know that, $n\left(B\cap B\right)=n\left(A\cap A\right)=1$, $n\left(A\cap B\right)=n\left(B\cap A\right)$

$\Rightarrow$$n\left((A\times B)\cap (B\times A)\right)=\left(3\times 1\times 3\times 1\right)$

$\Rightarrow$$n\left((A\times B)\cap (B\times A)\right)=9$

Thus, the value of $n\left((A\times B)\cap (B\times A)\right)$ is $9$.

Hence, option(B) i.e. $9$ is the correct answer.