If A and B are nonempty sets, prove that

Given: A = B, where A and B are nonempty sets.

Need to prove: $A \times B=B \times A$

Let us consider, $(x, y)^{\in}(A \times B)$

That means, $x \in_{A}$ and $y \in_{B}$

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

$\Rightarrow x^{\in}_{B}$ and $y^{\in} A$

$\Rightarrow(x, y)^{\in}(B \times A)$

That means, $(A \times B) \subseteq(B \times A) \cdots(1)$

Similarly we can prove,

$\Rightarrow(B \times A) \subseteq(A \times B) \cdots(2)$

So, by the definition of set we can say from (1) and (2),

A × B = B × A [Proved]