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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. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite. Formally, it states that for every indexed family ( S i ) i I {\displaystyle (S_{i})_{i\in I}} of nonempty sets there exists an indexed family ( x i ) i I {\displaystyle (x_{i})_{i\in I}} of elements such that x i S i {\displaystyle x_{i}\in S_{i}} for every i I {\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. MORE
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