korrespondenser

Korrespondenser is a term used in Danish-language mathematical literature to describe a relation that pairs elements of one set with elements of another. In English, the concept is typically called a relation or a correspondence. A korrespondenser between sets A and B can be described formally as a subset R of A × B. For each element a in A, there may be zero, one, or many elements b in B such that (a, b) is in R.

If every a in A is related to exactly one b in B, R is a function (also called a mapping). If, in addition, every b in B is related to exactly one a in A, R is a bijection. A korrespondenser may also be partial (not every a has a related b) or multivalued (an a relates to multiple b’s). The domain of R consists of those a in A for which there exists at least one related b, while the image (or range) is the set of all b in B that appear in a related pair.

Common terminology in English texts distinguishes between relations and functions, but many mathematical discussions view correspondenser as a general relation, with functions as a special, deterministic type. Related concepts include the graph of a relation, inverse relations, and composition of relations.

Examples: the identity relation on a set A is R = { (a, a) | a in A }, which is a bijective function. The relation “is a divisor of” on the positive integers is a multivalued, nonfunctional korrespondenser.

See also: relation, function, mapping, bijection, graph of a relation.