Relációra

Relációra is a Hungarian grammatical form of the noun reláció, which means relation. In mathematical and logical contexts the term reláció (relation) is central, and the form relációra appears in phrases such as "relációra vonatkozó definíció" (definition pertaining to the relation). The article here treats reláció as used in mathematics and its typical phrasing in Hungarian.

A relation on a set A is a subset R of the Cartesian product A×A; more generally, a relation between sets B and C is a subset R of B×C. A pair (a, b) belongs to R precisely when a is related to b by R. When R⊆A×A, R is called a binary relation on A. Relations can be described by listing pairs, by a rule, or by a graph.

Binary relations have several important properties. A relation is reflexive if every element relates to itself, symmetric if aRb implies bRa, antisymmetric if aRb and bRa imply a=b, and transitive if aRb and bRc imply aRc. A relation that is reflexive, symmetric, and transitive is an equivalence relation, which partitions the set into equivalence classes. A relation that is reflexive, antisymmetric, and transitive is a partial order, yielding a poset structure.

Relations can be composed: if R⊆A×B and S⊆B×C, then S∘R⊆A×C is a new relation. The inverse relation R−1 swaps the components of each pair. Relations can be represented as directed graphs, where vertices are elements of the base set and edges encode related pairs.

Applications span mathematics, logic, computer science, and databases (relational databases use the idea of relations). In Hungarian usage, the form relációra appears in constructions that specify the relation in question, illustrating the phrase-level role of the term within sentences.