Question

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

• 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 any given sets $P$ and $Q$, we know that $\left(P\times Q\right)\cap \left(Q\times P\right)=\left(P\cap Q\right)\times \left(Q\cap P\right)$

Therefore, $\left(A\times B\right)\cap \left(B\times A\right)=\left(A\cap B\right)\times \left(B\cap A\right)$

$\begin{array}{cccll}\therefore & n\left[\left(A\times B\right)\cap \left(B\times A\right)\right]& =& n\left[\left(A\cap B\right)\times \left(B\cap A\right)\right]& \\ & & =& n\left[A\cap B\right]\times n\left[B\cap A\right]& \left[\because n\left(P\times Q\right)=n\left(P\right)\times n\left(Q\right)\right]\\ & & =& n\left[A\cap B\right]\times n\left[A\cap B\right]& \left[\because \left(A\cap B\right)=\left(B\cap A\right)\right]\\ & & =& 3\times 3& \\ \Rightarrow & n\left[\left(A\times B\right)\cap \left(B\times A\right)\right]& =& 9& \end{array}$

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