disjunkt

Disjunkt, or disjoint in English, describes a relationship between two or more objects that do not share any elements. In mathematics and related fields, two sets A and B are disjunkt if their intersection is empty: A ∩ B = ∅. A collection of sets is called pairwise disjoint when every pair of distinct sets in the collection has empty intersection. A partition of a set S is a family of pairwise disjoint subsets whose union is S, effectively dividing S into non-overlapping blocks.

In practice, disjointness is a way to express that there is no overlap between the objects involved. The concept extends beyond sets to events in probability, propositions in logic, and categories in abstract mathematics. For example, two events A and B are disjoint if they cannot occur at the same time; this implies P(A ∪ B) = P(A) + P(B) for disjoint events, with generalizations to larger collections.

The term is used in several languages, including German, Norwegian, and Danish, where disjunkt denotes the same idea. In some contexts, “mutually exclusive” is used as a synonym, especially in probability and statistics, to emphasize that the occurrence of one outcome excludes the other.

A disjoint union is a construction that preserves the distinct identity of elements from each component, often denoted by ⊎. It is used to form coproducts in category theory and to define unions of disjoint sets in combinatorics and computer science.