Union

The union of two sets A and B, written A ∪ B, is the set of elements belonging to A, to B, or to both: A ∪ B = {x : x ∈ A or x ∈ B}. More generally, the union of a family of sets {Aᵢ}ᵢ∈I is ⋃ᵢ Aᵢ = {x : x ∈ Aᵢ for some i ∈ I}.

Union is commutative (A ∪ B = B ∪ A), associative ((A ∪ B) ∪ C = A ∪ (B ∪ C)), and idempotent (A ∪ A = A). The empty set is the identity: A ∪ ∅ = A. Together with intersection, union gives the power set of any set the structure of a Boolean algebra.

In a lattice, union corresponds to join (least upper bound). In the open sets of a topological space, arbitrary unions are allowed — this is one of the topology axioms — and the join of open sets is their union. In the Heyting algebra H of the semiotic universe, join (a ∨ b) is the semantic analogue of union: the weakest value that entails both a and b.