A symmetric relation is a type of binary relation. A homogeneous relation R {\displaystyle R} on a set X {\displaystyle X} is symmetric if:
where the notation a R b {\displaystyle aRb} means that the ordered pair ( a , b ) {\displaystyle (a,b)} is in the relation R {\displaystyle R} .
An example is the relation "is equal to", because if a = b is true then b = a is also true. 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.
"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"
"has coauthored a book with"
"is a homophone of"
"is a co-worker of"
"is a teammate of"
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.
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.
Note that S(n, k) refers to Stirling numbers of the second kind.
Commutative property – Property of some mathematical operations
Symmetry in mathematics
Symmetry – Mathematical invariance under transformations
