Question

If A and B are two non-empty sets such that A x B = B x A, show that A=B

Answer 1

Proof.
Notice that A x B consists of all pairs (a,b), where a belongs to A,
and b belongs to B, while
B x A consists of all pairs (b,a), where a
belongs to A, and b belongs to B.

Let a belongs to A.
We should prove
that a belongs to B as well.

Take any b from B.
Then
(a,b) belongs
to A x B = B x A,
whence
(a,b) = (b',a')
for some b' from B and a'
from A.
This means that

a=b' and b=a'.

Hence a belong to
B.
This proves that A is contained in B.
By similar argument B is contained
in A, and so A=B.