Axiom of union

The axiom of union asserts that every set of sets has a set containing exactly the elements of its members. It formalizes forming the union of a family of sets.

Union (set theory): The set of elements belonging to at least one member of a given family of sets. The axiom guarantees that this operation produces a set for every set-sized family.
Set (mathematics): A collection treated as a mathematical object, whose members are determined by membership. The axiom concerns a set whose members are themselves sets.
Transitive closure: The smallest transitive set containing a given set. Repeated unions generate the levels needed to collect everything reachable by membership.
Axiom of foundation: The axiom asserting that every nonempty set has an element disjoint from it. Foundation restricts membership structure, whereas union guarantees a set formed from existing members.
Axiom of pairing: The set-theoretic axiom asserting that any two sets belong together to some set. Together with union, pairing can form a set containing the elements of two chosen sets.
Membership (set theory): The relation stating that an object is an element of a set, written with the symbol ∈. The axiom specifies membership in the resulting set through membership in a member of the input set.
Ordinal number: A set representing the order type of a well-ordered set, in the von Neumann construction. For a limit ordinal, union combines all earlier ordinals into that ordinal.
Universal set: A set containing every set as an element. The union axiom forms unions of set-sized families, not a set containing all sets.
Axiom schema of replacement: An axiom schema asserting that definable functions map sets to sets. Replacement supplies sets of images; union then collects the elements of those images.
Empty set: The unique set with no elements. If the input family is empty, its union is the empty set.

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