Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a.

In mathematical notation, this is:

By definition, a 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").

Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if a = b) are actually independent of each other, as these examples show.

A symmetric relation that is also transitive and reflexive is an equivalence relation.

One way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edge's two vertices being the two entities so related. Thus, symmetric relations and undirected graphs are combinatorially equivalent objects.

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