Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
- Date: Thu, 02 Aug 2007 06:27:07 -0500
On Thu, 02 Aug 2007 09:26:17 GMT, Frederick Williams <"Frederick
Williams"@antispamhotmail.co.uk.invalid> wrote:
Scott wrote:
Hi:
I recently received a comment from someone that given |X| > 0 and |Y|
0, X x Y can be empty. Based upon what I know about set theory, thisdoesn't seem correct. Can someone confirm and deny this comment?
Thanks for you help in advance.
The axiom of choice implies that
X x Y = 0 iff (X = 0 or Y = 0) . . . . . . . . . . . . (*)
True but slightly silly and possibly misleading, since (*)
follows from ZF without AC.
Also
CartesianProduct{Y_i}_(i in I) = 0 iff (exists i)(Y_i = 0) if I =/= 0
Less important point: one might want to clarify exactly how
"A iff B if C" is to be parsed.
(i.e. the generalization of (*) to a non-empty family of sets) implies
the axiom of choice.
************************
David C. Ullrich
.
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