In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory. This axiom was introduced by Ernst Zermelo.

Informally, the axiom states that if is a set of sets, then the union of all sets in is still a set. In more basic terms, for each set there is a set whose elements are precisely the elements of the elements of .

Formal statement

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

<math display="block">\forall X\, \exists Y\, \forall u\, (u \in Y \leftrightarrow \exists z\, (u \in z \land z \in X))</math>

or in words:

:Given any set X, there is a set Y such that, for any element u, u is a member of Y if and only if there is a set z such that u is a member of z and z is a member of X.

or, more simply:

:For any set <math>X</math>, there is a set <math>\bigcup X</math> which consists of just the elements of the elements of that set <math>X</math>.

Consequences

The axiom of union allows one to unpack a set of sets and thus create a flatter set.

Together with the axiom of pairing, it implies that for any two sets and , their binary union is also a set.

Together with the axiom schema of replacement, the axiom of union implies that one can form the union of a family of sets indexed by a set.

The axiom of union is often used to construct the limit of an infinite sequence of sets . For example, it can be used to construct the supremum of any set of ordinal numbers.

Relation to Intersection

There is no corresponding axiom of intersection. If <math>A</math> is a nonempty set containing <math>E</math>, it is possible to form the intersection <math>\bigcap A</math> using the axiom schema of specification as

<math display="block">\bigcap A = \{c\in E:\forall D(D\in A\Rightarrow c\in D)\},</math>

so no separate axiom of intersection is necessary. (If A is the empty set, then trying to form the intersection of A as

:{c: 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 a universal set is antithetical to Zermelo–Fraenkel set theory.)

Notes

References

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Further reading

  • Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition).
  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. .

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