Statement:
M. Hall's Theorem: Let \(\{S(\alpha): \alpha\in A\}\) be a collection of finite subsets (of a set \(X\)) then if
(*) for each finite \(F \subseteq A\) there is an injective choice function on \(F\)
then there is an injective choice function on \(A\). (That is, a 1-1 function \(f\) such that \((\forall\alpha\in A)(f(\alpha)\in S(\alpha))\).) (According to a theorem of P. Hall (\(*\)) is equivalent to \(\left |\bigcup_{\alpha\in F} S(\alpha)\right|\ge |F|\). P. Hall's theorem does not require the axiom of choice.)
Howard_Rubin_Number:
107
Parameter(s):
This form does not depend on parameters
This form's transferability is:
Unknown
This form's negation transferability is:
Negation Transferable
Article Citations:
Hall-1948: Distinct representatives of subsets
Book references
Note connections:
The following forms are listed as conclusions of this form class in rfb1:
62,
96,
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