Suppose that A times B Phi where A and B are sets What can

Suppose that A times B = Phi, where A and B are sets. What can you conclude? Show that A times B notequalto B times A, where A and B are nonempty, unless A = B.

Solution

1) We can conclude that A = Ø or B = Ø. To prove this, suppose that neither A nor B were empty.

Then there would be elements a A and b B. This would give at last one element,namely (a, b), in A × B,

so A × B would not be the empty set. This contradiction shows thateither A or B (or both, it goes without saying) is empty.

2) if A = B, then
A x B = A x A = B x A

now suppose A B. That means that there\'s some element c such that either
c A & c B
or
c B & c A

  If it\'s the former, then choose any element b B, and then:
(c,b) A x B ... because c A & b B, but:
(c,b) B x A ... because c B
so that A x B B x A

And if it\'s the latter, then choose any element a A, and then:
(a,c) A x B ... because a A & c B, but:
(a,c) B x A ... because c A
so that A x B B x A