disjointness

Disjointness is a relation between two sets indicating they have no elements in common. Formally, sets A and B are disjoint if their intersection is empty, A ∩ B = ∅. By convention, a set is disjoint with the empty set, since ∅ ∩ A = ∅ for any A.

A collection of sets {A_i} is called pairwise disjoint (or mutually disjoint) if any two distinct members have empty intersection: for i ≠ j, A_i ∩ A_j = ∅. The property is preserved under taking subfamilies.

In probability and statistics, two events E and F are disjoint if they cannot occur together. If E and F are disjoint, then P(E ∪ F) = P(E) + P(F); in general, P(E ∪ F) ≤ P(E) + P(F).

In topology and analysis, disjointness describes subsets that share no points; for example, two disjoint open sets separate regions of a space. The concept also underpins partitions of spaces into disjoint pieces.

Disjointness has a standard construction in set theory: the disjoint union (coproduct) of a family of sets gathers the sets while keeping track of their origins, forming a larger set whose elements are pairs (i, x) with x ∈ A_i. This contrasts with a naive union, which may identify equal elements coming from different components.