Axiom of choice

Axiom of set theory

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one element chosen from each set, even if the collection is infinite. Formally, the axiom establishes existence rather than a construction; it states that for every set I {\displaystyle I} and every I {\displaystyle I} -indexed family ( S i) i ∈ I {\displaystyle (S_{i})_{i\in I}} of nonempty sets, there exists an I {\displaystyle I} -indexed set ( x i) i ∈ I {\displaystyle (x_{i})_{i\in I}} of elements of ∪ i ∈ I S i {\displaystyle \cup _{i\in I}S_{i}} such that x i ∈ S i {\displaystyle x_{i}\in S_{i}} for every i ∈ I {\displaystyle i\in I} .

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