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*);

- 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*.

*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.

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.