Question

prove that A×B=B ×A Which implies that A= B

Answer 1

If A−B=B−A
A−B=B−A then for any x∈A−B=B−Ax∈A−B=B−A we x∈A;x∈B;x∉A;x∉Bx∈A;x∈B;x∉A;x∉B.
That's a contradiction so A−B=B−AA−B=B−A is empty.

Thus there are no elements in AA that are not in BB.
In other words AA is a subset of BB.
Likewise there are no elements of BBthat are in AA. So BB is a subset of AA.

So A=BA=B.