In mathematical notation, this is:
- <math>\forall a, b \in X,\ a R b \Rightarrow \; b R a.</math>
- "is equal to" (equality) (whereas "is less than" is not symmetric)
- "is comparable to", for elements of a partially ordered set
- "... and ... are odd":
- "is married to" (in most legal systems)
- "is a fully biological sibling of"
- "is a homophone of"
Relationship to asymmetric and antisymmetric relations
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.
|Antisymmetric||equality||"is less than or equal to"|
|Not antisymmetric||congruence in modular arithmetic||"is divisible by", over the set of integers|
|Antisymmetric||"is the same person as, and is married"||"is the plural of"|
|Not antisymmetric||"is a full biological sibling of"||"preys on"|
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.