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The [[Axiom of choice|axiom of choice]] for an arbitrary (not necessarily disjoint) family of sets. E. Zermelo stated this axiom in 1904 in the form of the following assertion, which he called the principle of choice [[#References|[1]]]: For every family of non-empty sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099180/z0991801.png" /> one can choose from each of its terms exactly one representative and combine all these in a single set. He was the first to give a proof, based on his principle of choice, of his well-ordering theorem (cf. [[Zermelo theorem|Zermelo theorem]]). In 1906, B. Russell stated the axiom of choice in a multiplicative form: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099180/z0991802.png" /> is a disjoint set of non-empty sets, then the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099180/z0991803.png" /> is not empty. In 1908 Zermelo proved the equivalence of the multiplicative form of the axiom of choice and its usual statement.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Zermelo,   "Beweiss, dass jede Menge wohlgeordnet werden kann"  ''Math. Ann.'' , '''59'''  (1904)  pp. 514–516</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Fraenkel,   Y. Bar-Hillel,   "Foundations of set theory" , North-Holland  (1958)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Moore,   "Zermelo's axiom of choice" , Springer  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Rubin,   H. Rubin,   "Equivalents of the axiom of choice" , '''1–2''' , North-Holland  (1963–1985)</TD></TR></table>