dUnion

Dunion is a term used in mathematics and computer science to denote the disjoint union of a family of sets. Given a family of sets {X_i} indexed by i in I, the dunion ⊎_{i∈I} X_i is the set of pairs (i, x) with i∈I and x∈X_i. This construction separates elements coming from different X_i by tagging them with their index, ensuring that the same element value from different X_i's are treated as distinct elements in the dunion.

In set theory, this is also called the disjoint sum; in category theory, the coproduct. The term dunion emphasizes the disjointness property. Notation varies; some writers write ⊎X_i or ⊔ with tagging for disambiguation.

Example: X_1 = {a,b}, X_2 = {a,c}. The dunion is {(1,a),(1,b),(2,a),(2,c)}. If the X_i are already pairwise disjoint subsets of a common universe, the dunion is isomorphic to the ordinary union, with the tags providing disambiguation. Cardinality satisfies |⊎_{i∈I} X_i| = sum_{i∈I} |X_i|. The construction is associative and commutative up to natural isomorphism.

Applications include the design of sum types in programming languages, data schemas for heterogeneous records, and theoretical frameworks in algebra and topology. In programming, a tagged union or variant type implements a dunion by combining possibilities into one type with an explicit tag for the originating case. See also disjoint union, coproduct, sum type, and tagged union.