given: n (A) = 3 n (B) = 2

To prove: The sets A x B and B x A have an element in common if the sets A and B be two sets such that n (A) = 3 and n (B) = 2

Proof:

Case 1: No elements are common

Assuming:

A = (a, b, c) and B = (e, f)

So, we have:

A × B = {(a, e), (a, f), (b, e), (b, f), (c, e), (c, f)}

B × A = {(e, a), (e, b), (e, c), (f, a), (f, b), (f, c)}

There are no common ordered pair in A × B and B × A.

Case 2: One element is common

Assuming:

A = (a, b, c) and B = (a, f)

So, we have:

A × B = {(a, a), (a, f), (b, a), (b, f), (c, a), (c, f)}

B × A = {(a, a), (a, b), (a, c), (f, a), (f, b), (f, c)}

Here, A × B and B × A have one ordered pair in common.

Therefore, we can say that A × B and B × A will have elements in common if and only if sets A and B have an element in common.