The axiom of choice, also called Zermelo's axiom of choice, is an axiom of set theory asserting that for every family 
of nonempty sets,
there exists a function 
satisfying 
for all 
. The sets 
need not be disjoint. Zermelo
(1904, p. 516) formulated the principle in his proof
of the well ordering theorem.
An equivalent form asserts that, for every family of pairwise disjoint nonempty sets, there exists a set containing exactly one element in common with each member of the family. The axiom of choice is related to the first of Hilbert's problems.
In Zermelo-Fraenkel set theory (in the form omitting the axiom of choice), Zorn's lemma, the trichotomy law, and the well ordering theorem are equivalent to the axiom of choice (Mendelson 1997, p. 275). In contexts sensitive to the axiom of choice, the notation "ZF" is often used to denote Zermelo-Fraenkel without the axiom of choice, while "ZFC" is used if the axiom of choice is included.
In 1940, Gödel proved that the axiom of choice is consistent with the axioms of von Neumann-Bernays-Gödel set theory (a conservative extension of Zermelo-Fraenkel set theory). However, in 1963, Cohen (1963) unexpectedly demonstrated that the axiom of choice is also independent of Zermelo-Fraenkel set theory (Mendelson 1997; Boyer and Merzbacher 1991, pp. 610-611).
The Partition Principle asserts that every surjection 
between sets
implies the existence of an injection 
, without requiring the injection
to choose a point from each fiber.
