Question

If A and B are two sets having 3 elements in common.If n$\left(A\right)=5,n\left(B\right)=4$, find $n(A\times B)$ and $n[(A\times B)\cap (B\times A)].$

Answer 1

Let $A=\{a,b,c,d,e\}$

and $B=\{1,2,3,4\}$

$A\times B=(a,1),(a,2),(a,3),(a,4),(b,1),(b,2),(b,3),(b,4),(c,1),(c,2),(c,3),(c,4),(d,1),(d,2),(d,3),(d,4),(e,1),(e,2),(e,3),(e,4)$

$\therefore n(A\times B)=n\left(A\right)n\left(B\right)=20$

We know that $(A\times B)\cap (B\times A)=(A\cap B)\times (B\cap A)$

So,$(A\times B)\cap (B\times A)=(A\cap B)\times (B\cap A)$

Given:Common elements of $A$ and $B=3$

$\therefore n(A\cap B)=3$

So,$n[(A\cap B)\times (B\cap A)]=n(A\cap B)n(B\cap A)=3\times 3=9$ elements

$\therefore n[(A\cap B)\times (B\cap A)]=9$ elements