Let A and B be two non-empty sets having n elements in common. Then, the number of elements common to $A×B$ and $B×A$ is
Answer & explanation
Correct answer: option 3
We know that
$(A × B)∩(C × D) = (A∩C) × (B∩D)$
$∴(A × B)∩(B × A) = (A∩B) × (B∩A)$
$⇒(A × B)∩(B × A) = (A∩B) × (A∩B)$
It is given that $A∩B$ has n elements.
$∴(A∩B) × (A∩B)$ has $n^2$ elements.
But, $(A× B)∩(B × A) = (A∩B) × (A∩B)$
$∴(A× B)∩(B × A)$ has $n^2$ elements in common.