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If A and B are nonempty sets, prove that A × B = B × A ⇔ A = B

Answer : Given: A = B, where A and B are nonempty sets. Need to prove: A × B = B × A

Let us consider, (x, y)    (A × B)

That means, x    A and y    B

As given in the problem A = B, we can write,

⇒ x     B and y    A

⇒ (x, y)    (B × A)

That means, (A × B) ⊆ (B × A)—- (1)

Similarly we can prove,

⇒ (B × A) ⊆ (A × B)—- (2)

So, by the definition of set we can say from (1) and (2), A × B = B × A [Proved]