Axiom of Choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. It states that for every indexed family ${\displaystyle (S_{i})_{i\in I}}$ of nonempty sets there exists an indexed family ${\displaystyle (x_{i})_{i\in I}}$ of elements such that ${\displaystyle x_{i}\in S_{i}}$ for every ${\displaystyle i\in I}$. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

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