Axiom of union

In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–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. It was introduced by Zermelo (1908). Together with the axiom of pairing this implies that for any two sets, there is a set (called their union) that contains exactly the elements of both. Together with the axiom of replacement the axiom of union implies that one can form the union of a family of sets indexed by a set.

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

or in words:

or, more simply:

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 ${\displaystyle \bigcup A}$. Thus the essence of the axiom is:

Note that the union of A and B, commonly written as ${\displaystyle A\cup B}$, can be written as ${\displaystyle \bigcup \{A,B\}}$. Thus, the ordinary construction of unions is trivial given the axiom of pairing.

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