relationen

Relationen, the plural of the German term for relation, denote the concept of connections or associations studied across disciplines. In mathematics and logic, a relation on a set A is a subset of the Cartesian product A^n; most commonly one considers binary relations, a subset of A×A, describing when two elements are in a given relation. If a and b are related by R, we write a R b. Examples include equality (a = b), order (a < b) on numbers, and divisibility (a | b). Relations can be analyzed by properties such as reflexivity (every a R a), symmetry (a R b implies b R a), and transitivity (a R b and b R c imply a R c). Antisymmetry, equivalence relations (reflexive, symmetric, transitive), and partial orders (reflexive, antisymmetric, transitive) are important special cases. A relation can also have a converse and a composition with other relations, and it may be represented as a directed graph with elements as vertices and related pairs as edges.

In computer science and information systems, a relation often refers to a database table in the relational model introduced by E. F. Codd. A relation consists of a set of tuples, each matching a fixed schema of attributes. Operations such as selection, projection, and join manipulate relations, and keys enforce unique identification of tuples while referential integrity links related tables.

Beyond math and databases, relations are used in linguistics to describe semantic roles, in philosophy to discuss relational properties, and in graph theory where a relation corresponds to directed edges. A function is a special kind of relation with a unique output for each input.