Symmetric relation
A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if a = b is true then b = a is also true. Formally, a binary relation R over a set X is symmetric if:
If RT represents the converse of R, then R is symmetric if and only if R = RT.
Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.
Examples
In mathematics
- "is equal to"
- "is comparable to", for elements of a partially ordered set
- "... and... are odd":
Outside mathematics
- "is married to"
- "is a fully biological sibling of"
- "is a homophone of"
- "is co-worker of"
- "is teammate of"
Relationship to asymmetric and antisymmetric relations
Symmetric and antisymmetric are actually independent of each other, as these examples show.
| Symmetric | Not symmetric | |
| Antisymmetric | equality | "is less than or equal to" |
| Not antisymmetric | congruence in modular arithmetic | "is divisible by", over the set of integers |
| Symmetric | Not symmetric | |
| Antisymmetric | "is the same person as, and is married" | "is the plural of" |
| Not antisymmetric | "is a full biological sibling of" | "preys on" |
Properties
- A symmetric and transitive relation is always quasireflexive.
- A symmetric, transitive, and reflexive relation is called 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.