product of non-empty set of non-empty sets is non-empty
In this entry, we show that the statement:
(*) the non-empty generalized cartesian product of non-empty sets is non-empty
AC implies (*).
Suppose is a set of non-empty sets, with . We want to show that
is non-empty. Let . Then, by AC, there is a function such that for every . Define by . Then as a result, is non-empty. ∎
Remark. The statement that if , then implies does not require AC: if is non-empty, then there is a function , and, as , , which means , or that for some .
(*) implies AC.
Suppose is a set of non-empty sets. If itself is empty, then the choice function is the empty set. So suppose that is non-empty. We want to find a (choice) function , such that for every . Index elements of by itself: for each . So by assumption. Hence, by (*), the (non-empty) cartesian product of the is non-empty. But an element of is just a function whose domain is and whose codomain is the union of the , or , such that , which is precisely . ∎
|Title||product of non-empty set of non-empty sets is non-empty|
|Date of creation||2013-03-22 18:44:28|
|Last modified on||2013-03-22 18:44:28|
|Last modified by||CWoo (3771)|