Axiom of Choice

The axiom of choice (AC) states:

For every indexed family $(A_i)_{i\in I}$ of nonempty sets , there exists a function $c\colon I\to \bigcup_{i\in I} A_i$ (see union ) such that $c(i)\in A_i$ for every $i\in I$. Such a function $c$ is called a choice function for the family.

AC is independent of the other ZFC axioms (ZF) and is equivalent (over ZF) to several major principles, including Zorn's lemma and the well-ordering theorem .