Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?

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, this
doesn'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) . . . . . . . . . . . . (*)

Also

CartesianProduct{Y_i}_(i in I) = 0 iff (exists i)(Y_i = 0) if I =/= 0

(i.e. the generalization of (*) to a non-empty family of sets) implies
the axiom of choice.

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