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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").
Up to a relation by a rigid motion, they are equal if related by a direct isometry. Isometries have been used to unify the working definition of symmetry in geometry and for functions, probability distributions, matrices, strings, graphs, etc. [7]
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
Peer relationships, such as can be governed by the Golden Rule, are based on symmetry, whereas power relationships are based on asymmetry. [35] Symmetrical relationships can to some degree be maintained by simple (game theory) strategies seen in symmetric games such as tit for tat. [36]
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.
For example, if the function f is defined as (,) = + then is a symmetric function. For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then a R b ⇔ b R a {\displaystyle aRb\Leftrightarrow bRa} .
A relation is connex if and only if its complement is asymmetric. A non-example is the "less than or equal" relation ≤ {\displaystyle \leq } . This is not asymmetric, because reversing for example, x ≤ x {\displaystyle x\leq x} produces x ≤ x {\displaystyle x\leq x} and both are true.
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.