disjoint

Disjoint describes a relationship between two sets or subobjects in which they share no elements. In set theory, two sets A and B are disjoint if their intersection is empty: A ∩ B = ∅. A collection {A_i} is called pairwise disjoint (or mutually disjoint) if A_i ∩ A_j = ∅ for all i ≠ j. When combining disjoint pieces, the resulting union is often referred to as a disjoint union, sometimes denoted by ⊔, emphasizing that the pieces contribute without overlap.

Examples illustrate the concept: {1, 2} and {3, 4} are disjoint, while [0, 1] and [1, 2] are not, since they share the point 1. A disjoint union of sets yields a larger set whose elements come from exactly one of the parts.

In probability, disjoint events (also called mutually exclusive) cannot occur at the same time. If A and B are disjoint, then P(A ∪ B) = P(A) + P(B). If they are not disjoint, inclusion-exclusion gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The term has broader use across mathematics and related fields, where disjointness often expresses a lack of overlap.

Beyond basic set theory, disjointness appears in topology (disjoint subspaces), graph theory (vertex-disjoint or edge-disjoint subgraphs), and computer science (disjoint-set data structures for maintaining a partition of a universe into disjoint blocks).

Overall, disjointness is a fundamental way to formalize the idea that two objects do not share any elements.