Let A and B be nonempty sets Prove that A B B A iff A BS

Let A and B be nonempty sets. Prove that A × B = B × A iff A = B.

Solution

Let\'s take A x B = B x A and we need to show that A = B.

So let us take x € A and y € B then (x,y) € A x B

And we have hypothesis A x B = B x A so (x,y) € B x A also.

Then x is in B and y is in A.

So x is in A and B also so A is subset to B

and y is in B and A also so B is subset to A.

when A is subset to B and B is subset to A then A = B.

Now let us take A = B and prove that A x B = B x A

Then lets take (x,y) € A x B and x € A and y € B.

and we have A = B so obviosly x € B and y € A.

so (x,y) € B x A

A x B is a subset of B x A

Then x is in B and y is in A. Then x is in A and y is in B since A = B

so (x, y) € A x B.

Thus, B x A is a subset of A x B

so when A x B Is subset B x A and B x A is subset of A x B then A x B = B x A.

so A × B = B × A iff A = B