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A relation is asymmetric if, and only if, it is antisymmetric and irreflexive. A transitive relation is irreflexive if, and only if, it is asymmetric. A relation is reflexive if, and only if, its complement is irreflexive. A relation is strongly connected if, and only if, it is connected and reflexive.
A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty domain).
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.. More precisely, if , …, are elements of a (left) module M over a ring R (the case of a vector space over a field is a special case), a relation between , …, is a sequence (, …,) of elements of R such that
In the particular case of groups, congruence relations can be described in elementary terms as follows: If G is a group (with identity element e and operation *) and ~ is a binary relation on G, then ~ is a congruence whenever: Given any element a of G, a ~ a (reflexivity); Given any elements a and b of G, if a ~ b, then b ~ a ;
Depending on authors, the term "maps" or the term "functions" may be reserved for specific kinds of functions or morphisms (e.g., function as an analytic term and map as a general term). mathematics See mathematics. multivalued A "multivalued function” from a set A to a set B is a function from A to the subsets of B.
In mathematics, a binary relation R ⊆ X×Y between two sets X and Y is total (or left total) if the source set X equals the domain {x : there is a y with xRy}.Conversely, R is called right total if Y equals the range {y : there is an x with xRy}.
A relation is transitive if it is closed under this operation, and the transitive closure of a relation is its closure under this operation. A preorder is a relation that is reflective and transitive. It follows that the reflexive transitive closure of a relation is the smallest preorder containing it
An example of an irreflexive relation, which means that it does not relate any element to itself, is the "greater than" relation (>) on the real numbers. Not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but others are not (that is, neither all nor none are).
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