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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").
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.
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
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.
The edge relation [note 1] of a tournament graph is always a connected relation on the set of ' s vertices. If a strongly connected relation is symmetric, it is the universal relation. A relation is strongly connected if, and only if, it is connected and reflexive. [proof 1]
The resulting relation is reflexive since the preorder is reflexive; transitive by applying the transitivity of twice; and symmetric by definition. Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ∼ , {\displaystyle S/\sim ,} which is the set of all equivalence classes of ∼ ...
Visualization of relations leans on graph theory: For relations on a set (homogeneous relations), a directed graph illustrates a relation and a graph a symmetric relation. For heterogeneous relations a hypergraph has edges possibly with more than two nodes, and can be illustrated by a bipartite graph .
A dyadic relation R = S × S is symmetric if for all elements a, b in S, whenever it is true that Rab, it is also true that Rba. [13] Thus, the relation "is the same age as" is symmetric, for if Paul is the same age as Mary, then Mary is the same age as Paul.