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Functional and not injective. For example, the red relation in the diagram is many-to-one, but the green, blue and black ones are not. Many-to-many [d] Not injective nor functional. For example, the black relation in the diagram is many-to-many, but the red, green and blue ones are not. Uniqueness and totality properties: A function [d]
For example, ≥ is a reflexive relation but > is not. Irreflexive (or strict) for all x ∈ X, not xRx. For example, > is an irreflexive relation, but ≥ is not. Coreflexive for all x, y ∈ X, if xRy then x = y. [7] For example, the relation over the integers in which each odd number is related to itself is a coreflexive relation.
For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle ...
For example, the green relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function. A surjection: a function that is surjective. For example, the green relation in the diagram is a surjection, but the red one is not. A bijection: a function that is injective and surjective.
The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The canonical map ker : X^X → Con X, relates the monoid X^X of all functions on X and Con X. ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f : X → X to its ...
David Rydeheard and Rod Burstall consider Rel to have objects that are homogeneous relations. For example, A is a set and R ⊆ A × A is a binary relation on A.The morphisms of this category are functions between sets that preserve a relation: Say S ⊆ B × B is a second relation and f: A → B is a function such that () (), then f is a morphism.
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] The ...
This notion of "total" should not be confused with that of a total relation in the sense that for all there is a so that (see serial relation). Connectedness features prominently in the definition of total orders : a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected.