Disjunktsed

Disjunktsed is an Estonian adjective meaning disjoint or mutually exclusive. In mathematical and scientific usage, it describes collections of objects that do not overlap or share common elements.

In set theory, a family of sets {A_i} is disjoint if the intersection of any two distinct members is empty: A_i ∩ A_j = ∅ for i ≠ j. A pair of disjoint sets have no elements in common, and this property can extend to finite or infinite families. Disjointness is often used to simplify counting, probability, and decomposition arguments.

In probability theory, two events A and B are disjoint (mutually exclusive) if they cannot occur at the same time, so A ∩ B = ∅ and P(A ∪ B) = P(A) + P(B). This concept generalizes to collections of events and underlies the calculation of probabilities for unions of disjoint events.

The disjoint union is a construction that combines multiple sets into one while preserving the identity of each component. It is typically denoted as a⊔A_i or described as the union of disjoint copies of the sets, ensuring that elements from different components remain distinguishable within the new set.

In logic and linguistics, disjointness relates to disjunctions, including exclusive or (XOR) in certain contexts. The term disjunktsed may appear in discussions of logical forms or grammatical structures where alternatives are presented as separate, non-overlapping options.

Examples illustrate the concept: A = {1, 2} and B = {3, 4} are disjoint since A ∩ B = ∅. The events “rolling a 1” and “rolling a 2” on a single die are disjoint. Intervals [0, 1] and [1, 2] are not disjoint because they share the point 1, whereas [0, 1) and (1, 2] are disjoint.

Etymology traces the term to Latin roots through European mathematical vocabulary and is standard in Estonian mathematical and technical literature. See also disjoint set, disjunction, union, probability.