Axiom of Choice
# Definition
## Definition
Axiom of Choice (AC) : Every family of nonempty sets has a choice function.
## Formally
$ X $ : an arbitrary set
$ \mathscr{F} $ : a family of $ S \ ;\ S \subset X $
### Axiom of Choice :
$$
\forall \mathscr{F}.\
\left[
\begin{array}{}
\emptyset \notin \mathscr{F}\\
\ \\
\hline \\
\forall S \in \mathscr{F}. \exists f : \mathscr{F} \rightarrow \cup_{S \in \mathscr{F}} S.\quad
f(S) \in S
\end{array}
\right]
$$
### Another definition :
$$
\forall \mathscr{F}. \
\left[
\begin{array}{}
\emptyset \notin \mathscr{F}\\
\ \\
\hline\\
\prod_{S \in \mathscr{F}} S \neq \emptyset
\end{array}
\right]
$$