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Axiom of choice - Statement

Axiom of choice - Statement: Encyclopedia II - Axiom of choice - Statement

The axiom of choice states: Let X be a set of non-empty sets. Then we can choose a member from each set in X. Stated more formally: Let X be a set of non-empty sets. Then there exists a choice function f defined on X. In other words, there exists a function f defined on X, such that for each set s in X, f(s) is an element of s. Another formulation of the axiom of choice states: Given any set of mutually disjoint non-empty sets, there exists at least one set that contains exactly one eleme ...

See also:

Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

Axiom of choice, Axiom of choice - Independence of AC, Axiom of choice - Quotes, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Results requiring ¬AC, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Weaker forms of AC

Axiom of choice: Encyclopedia II - Axiom of choice - Statement

Axiom of choice - Statement

The axiom of choice states:

Let X be a set of non-empty sets. Then we can choose a member from each set in X.

Stated more formally:

Let X be a set of non-empty sets. Then there exists a choice function f defined on X. In other words, there exists a function f defined on X, such that for each set s in X, f(s) is an element of s.

Another formulation of the axiom of choice states:

Given any set of mutually disjoint non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets.

Other related archives

1904, A. K. Dewdney, Algebra, April Fool's Day, Axiom of dependent choice, Baire category theorem, Banach-Alaoglu theorem, Banach–Tarski paradox, Bertrand Russell, Boolean prime ideal theorem, Ernst Zermelo, Functional analysis, General topology, Hahn-Banach theorem, Hausdorff paradox, Hilbert space, Kurt Gödel, Measure theory, Nielsen-Schreier theorem, Paul Cohen, Per Martin-Löf, Scientific American, Set theory, Set-builder notation, Stone's representation theorem for Boolean algebras, Stone-Cech compactification, Trichotomy, Tychonoff space, Tychonoff's theorem, Vitali theorem, Zermelo-Fraenkel set theory, Zorn's lemma, algebraic closure, axiom, axiom of countable choice, axiom of dependent choice, axiom of uniformization, basis, cardinality, category, choice function, choose, closed, closed graph theorem, closure, compact, compactness, complete, constructive logic, constructivists, countably many, determined, field, field extension, functional analysis, game, generalized continuum hypothesis, infinite, infinitely, injection, intuitionistic logic, intuitionistic type theory, law of excluded middle, linear functionals, logically independent, mathematical induction, mathematics, maximal ideal, measurable, metric spaces, model, natural numbers, negation, non-empty, non-measurable sets, nonconstructive, open, open mapping theorem, product, real analysis, real numbers, ring, set, set theory, skeleton, topological spaces, transcendence basis, ultrafilter lemma, uniform spaces, union, vector space, well-ordered, well-ordering principle, well-ordering theorem

Adapted from the Wikipedia article "Statement", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki

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