MappingRelationen

MappingRelationen are binary relations that encode a mapping from one set to another. In practice, they are often used to represent the graph of a function, capturing the pairs (a, b) where b is the image of a under the mapping.

Formal description typically treats MappingRelationen as subsets R of the Cartesian product A × B, where A is the source set and B is the target set. If for every a in A there exists at most one b in B with (a, b) in R, the relation is the graph of a partial function from A to B. If every a ∈ A has a unique image, R is the graph of a total function A → B. The inverse relation R⁻¹, when defined, is a relation from B to A representing preimages.

Key properties depend on the underlying mapping. When R is functional, it may be injective (distinct a map to distinct b) or surjective (every b has at least one preimage). If R is not functional, a single a may relate to multiple b values, modeling non-deterministic mappings or multi-valued correspondences.

Applications of MappingRelationen span several fields. In mathematics and computer science, they formalize the concept of a function as a relation. In databases and data integration, mappings between schemas or ontologies are often expressed as such relations. They also appear in programming language semantics, where the behavior of functions can be described relationally, and in formal analyses of data transformations and type systems.

See also: relation, function, graph of a function, relational composition, inverse relation.