Axiom of union

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo-Fraenkel set theory, stating that, for any two sets, there is a set that contains the exactly elements of both.

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

∀ A, ∃ B, ∀ C, C ∈ B ↔ (∃ D, D ∈ A ∧ C ∈ D);

or in words:

Given any set A, there is a set B such that, given any set C, C is a member of B if and only if there is a set D such that D is a member of A and C is a member of D.

To understand this axiom, note that the clause involving D in the symbolic statement above states that C is a member of some member of A. Thus, 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. We can use the axiom of extensionality to show that this set B is unique. We call the set B the union of A, and denote it ∪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. In the case where A is the empty set, there is no intersection of A in Zermelo-Fraenkel set theory. On the other hand, if A has some member B, then we can form the intersection ∩A as {C in B : for all D in A, C is in D} using the axiom schema of specification.