Question

If A and B are nonempty sets, prove that 

A × B = B × A ⇔ A = B

Answer 1

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]