Let A and B be two sets such that the Cartesian product $$A \times B$$ consists of four elements. If two elements of $$A \times B$$ are (1, 4) and (4, 1), then

It is given that in total A x B has 4 elements in it. This statement implies that there are possibility of {n(A) = 2, n(B) = 2} & {n(A) = 1, n(B) = 4} . But it is given that (1,4) & (4,1) are 2 elements of A x B, this implies that A must have two different elements that are {1,4} . Hence, we can say that n(A) = 2 & n(B) = 2.

As we deduced that A = {1,4} , lets find the set B which has 2 elements in it :

The given elements also mandates there should be {4,1} in set B, because then only the A x B can have (1,4) & (4,1).

Hence set A = {1,4} & set B = {4,1} . This is nothing but A = B. (since both sets are same)

As A = B, we can definitely say that A x B = B x A.