Question:
If $A$ and $B$ be two sets such that $n(A)=3, n(B)=4$ and $n(A \cap B)=2$ then find.
(i) $n(A \times B)$
(ii) $n(B \times A)$
(iii) $n(A \times B) \cap(B \times A)$
Solution:
Given: $n(A)=3, n(B)=4$ and $n(A \cap B)=2$
(i) $n(A \times B)=n(A) \times n(B)$
$\Rightarrow n(A \times B)=3 \times 4$
$\Rightarrow n(A \times B)=12$
(ii) $n(B \times A)=n(B) \times n(A)$
$\Rightarrow n(B \times A)=4 \times 3$
$\Rightarrow \mathrm{n}(\mathrm{B} \times \mathrm{A})=12$
(iii) $n((A \times B) \cap(B \times A))=n(A \times B)+n(B \times A)-n((A \times B) \cup(B \times A))$
$n((A \times B) \cap(B \times A))=n(A \times B)+n(B \times A)-n(A \times B)+n(B \times A)$
$n((A \times B) \cap(B \times A))=0$