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A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite. A preorder is reflexive and transitive. A congruence relation is an equivalence relation whose domain X {\displaystyle X} is also the underlying set for an algebraic structure , and which respects the additional structure.
Symmetric and antisymmetric relations. By definition, a nonempty relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be related to a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on").
Relations that satisfy certain combinations of the above properties are particularly useful, and thus have received names by their own. Equivalence relation A relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity. Orderings: Partial order
Within the framework of Zermelo–Fraenkel set theory, relations are by definition restricted to sets (a binary relation on a set A is a subset of the Cartesian product A × A), and there is no set of all sets in Zermelo–Fraenkel set theory.
A relation can be both symmetric and antisymmetric (in this case, it must be coreflexive), and there are relations which are neither symmetric nor antisymmetric (for example, the "preys on" relation on biological species). Antisymmetry is different from asymmetry: a relation is asymmetric if and only if it is antisymmetric and irreflexive.
The edges of a graph define a symmetric relation on the vertices, called the adjacency relation. Specifically, two vertices x and y are adjacent if {x, y} is an edge. A graph is fully determined by its adjacency matrix A, which is an n × n square matrix, with A ij specifying the number of connections from vertex i to vertex j.
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.
The root system of the exceptional Lie group E 8.Lie groups have many symmetries. Symmetry occurs not only in geometry, but also in other branches of mathematics.Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.