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:
 * $$\forall a, b \in X(a R b \Leftrightarrow b R a) ,$$

where the notation aRb means that (a, b) ∈ R.

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.

In mathematics

 * "is equal to" (equality) (whereas "is less than" is not symmetric)
 * "is comparable to", for elements of a partially ordered set
 * "... and ... are odd":
 * [[Image:Bothodd.png]]

Outside mathematics

 * "is married to" (in most legal systems)
 * "is a fully biological sibling of"
 * "is a homophone of"
 * "is a co-worker of"
 * "is a 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.

Properties

 * A symmetric and transitive relation is always quasireflexive.
 * One way to count the symmetric relations on n elements, that in their binary matrix representation the upper right triangle determines the relation fully, and it can be arbitrary given, thus there are as many symmetric relations as n × n binary upper triangle matrices, 2n(n+1)/2.