For any two sets A and B, show that

We know

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

Here $A$ and $B$ have an element in common i.e., $n(A \cap B)=1=(B \cap A)$

So, $n((A \times B) \cap(B \times A))=n((A \cap B) \times(B \cap A))=n(A \cap B) \times n(B \cap A)=1 \times 1=1$

That means, $A \times B$ and $B \times A$ have an element in common if and only if $A$ and $B$ have an element in common. [Proved]