Question

Show that A x B does not equal B x A, when A and B are nonempty unless A=B.

Answer 1

AXB= a E A and b E B where a is in A and b is in B

BXA= b E B and a E A

Since Cartesian product is a set of ordered pairs, ordering is important hence (a, b) does not equal to (b, a) unless a and b are equal

Explanation:

Answer 2

We will assume A×B = B×A and show that A and B are necessarily the same.

Assume A×B = B×A. Let a ∈ A and b ∈ B. Then (a,b) ∈ A×B. Since A×B = B×A, we have (a,b) ∈ B×A. That is, a ∈ B and b ∈ A. Therefore, a ∈ A implies a ∈ B and b ∈ B implies b ∈ A. So A = B, as we had set out to do.