Paarweise

Paarweise is a German adverb meaning in pairs or two at a time. It is used to describe actions, relations, or properties that involve elements grouped in twos, or to characterize a set of objects by their pairwise relations. The term derives from the noun Paar (pair) and the suffix -weise, which forms adverbial expressions of manner or mode.

In mathematics and probability, paarweise appears in common phrases such as paarweise verschieden and paarweise unabhängig. A sequence or set {a1, a2, ..., an} is paarweise verschieden if ai ≠ aj for all i ≠ j, i.e., all elements are distinct pairwise. A collection of events E1, E2, ..., En is paarweise unabhängig if for every i ≠ j, P(Ei ∩ Ej) = P(Ei)P(Ej). This is weaker than mutual (or total) independence, which requires independence for all finite intersections, not just pairs.

In combinatorics and related fields, paarweise considerations often arise when analyzing pairings, matchings, or partitions into two-element blocks. For example, a partition of a set into pairs is a way of pairing elements, while a pairwise disjoint family consists of subsets that do not share elements pairwise.

The term also appears in data analysis and decision theory, where pairwise comparisons are used to rank options, assess similarities, or build preference models. Here, decisions or evaluations are made by examining elements two at a time rather than collectively.

See also: pair, pairwise comparison, pairwise independence, mutual independence, pairwise distinct.