This article needs additional citations for verification. (July 2021)
|Transitive Binary Relations|
|All definitions tacitly require the homogeneous relation be transitive: |
A "✓" indicates that the column property is required in the row definition.
For example, the definition of an equivalence relation requires it to be symmetric.
Listed here are additional properties that a homogeneous relation may satisfy.
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:
- "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 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").
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.