# Axiom of union

In

axiomatic set theory and the branches oflogic ,mathematics , andcomputer science that use it, the**axiom of union**is one of theaxiom s ofZermelo-Fraenkel set theory , stating that, for any set "x" there is a set "y" whose elements are precisely the elements of the elements of "x". Together with theaxiom of pairing this implies that for any two sets, there is a set that contains exactly the elements of both.**Formal statement**In the

formal language of the Zermelo-Fraenkel axioms, the axiom reads::$forall\; A,\; exist\; B,\; forall\; c,\; (c\; in\; B\; iff\; exist\; D,\; (c\; in\; D\; and\; D\; in\; A),)$or in words::Given any set "A", there is a set "B" such that, for any element "c", "c" is a member of "B"if and only if there is a set "D" such that "c" is a member of "D" and "D" is a member of "A".**Interpretation**What the axiom is really saying is that, given a set "A", we can find a set "B" whose members are precisely the members of the members of "A". By the

axiom of extensionality this set "B" is unique and it is called the "union" of "A", and denoted ∪"A". Thus the essence of the axiom is::The union of a set is a set.The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternative

axiomatization of set theory.Note that there is no corresponding axiom of intersection. If "A" is a "nonempty" set, then we can form the intersection ∩"A" using the

axiom schema of specification as {"c" in "B" : for all "D" in "A", "c" is in "D"}, so no separate axiom of intersection is necessary. (If "A" is theempty set , then trying to form the intersection of "A" as {"c" such that for all "D" in "A", "c" is in "D"} is not permitted by the axioms. Moreover, if such a set existed, then it would contain every set in the "universe", but the notion of universe is antithetical to Zermelo-Fraenkel set theory.)**References***

Paul Halmos , "Naive set theory". Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).

*Jech, Thomas, 2003. "Set Theory: The Third Millennium Edition, Revised and Expanded". Springer. ISBN 3-540-44085-2.

*Kunen, Kenneth, 1980. "Set Theory: An Introduction to Independence Proofs". Elsevier. ISBN 0-444-86839-9.**External links***planetmath reference|id=4394|title=Axiom of Union

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