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.
- 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.