# Axiom of choice

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The axiom states that if ${\displaystyle {\mathcal {A}}}$ is a family of non-empty sets, there is a choice function ${\displaystyle f:{\mathcal {A}}\rightarrow \cup {\mathcal {A}}}$ such that for each ${\displaystyle A\in {\mathcal {A}}}$ we have ${\displaystyle f(A)\in A}$: that is, ${\displaystyle f}$ "chooses" an element of each member of the family ${\displaystyle {\mathcal {A}}}$.