This article needs additional citations for verification. (July 2021)
|Transitive binary relations|
|homogeneous relation be transitive: for all if and then and there are additional properties that a homogeneous relation may satisfy.indicates that the column's property is required by the definition of the row's term (at the very left). For example, the definition of an equivalence relation requires it to be symmetric. All definitions tacitly require the|
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:
where the notation means that .
- "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"
- "is co-worker of"
- "is teammate of"
Relationship to asymmetric and antisymmetric relationsEdit
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").
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.