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Confluence (term rewriting), discusses several unusual but fundamental properties of binary relations; Correspondence (algebraic geometry), a binary relation defined by algebraic equations; Hasse diagram, a graphic means to display an order relation; Incidence structure, a heterogeneous relation between set of points and lines
Pages in category "Properties of binary relations" The following 22 pages are in this category, out of 22 total. This list may not reflect recent changes. A.
Various properties of relations are investigated. A relation R is reflexive if xRx holds for all x, and irreflexive if xRx holds for no x. It is symmetric if xRy always implies yRx, and asymmetric if xRy implies that yRx is impossible. It is transitive if xRy and yRz always implies xRz.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
This property makes the set of all binary relations on a set a semigroup with involution. The composition of (partial) functions (that is, functional relations) is again a (partial) function. If R {\displaystyle R} and S {\displaystyle S} are injective , then R ; S {\displaystyle R\mathbin {;} S} is injective, which conversely implies only the ...
A relation R is called intransitive if it is not transitive, that is, if xRy and yRz, but not xRz, for some x, y, z. In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold. For example, the relation defined by xRy if xy is an even number is intransitive, [13] but not antitransitive. [14]
Implications and conflicts between properties of homogeneous binary relations Implications (blue) and conflicts (red) between properties (yellow) of homogeneous binary relations. For example, every asymmetric relation is irreflexive (" ASym ⇒ Irrefl "), and no relation on a non-empty set can be both irreflexive and reflexive (" Irrefl # Refl ").
The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of ) is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given any binary relation R on X , the equivalence relation generated by R is the intersection of all equivalence relations containing R ...