proof that a relation is union of functions if and only if AC
Suppose that is a relation with . Let be given
by . There is be a choice function
on . Let , and for each pair
, let send to and agree
with elsewhere. Let . Clearly , so suppose
; then there is a pair such that . Either , or . In each case, .
Thus, For each pair , , so . Therefore, .
Suppose that is set of nonempty sets. Let . A set is an element of if and only if for some . Thus, . There is a set of functions, each of which has domain , such that . Let ; then , and for each pair , ; i.e., . Each such is, thus, a choice function on . ∎
|Title||proof that a relation is union of functions if and only if AC|
|Date of creation||2013-03-22 16:34:23|
|Last modified on||2013-03-22 16:34:23|
|Last modified by||ratboy (4018)|