Axiom of choice
Variants

There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.

One variation avoids the use of choice functions by, in effect, replacing each choice function with its range:

This can be formalized in first-order logic as:

Note that P ∨ Q ∨ R is logically equivalent to (¬P ∧ ¬Q) → R. In English, this first-order sentence reads:

This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.

Another equivalent axiom only considers collections X that are essentially powersets of other sets:

Authors who use this formulation often speak of the choice function on A, but this is a slightly different notion of choice function. Its domain is the power set of A (with the empty set removed), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets. With this alternate notion of choice function, the axiom of choice can be compactly stated as

which is equivalent to

The negation of the axiom can thus be expressed as:
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