Symmetric relation

A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if:

${\displaystyle \forall a,b\in X(aRb\Leftrightarrow bRa),}$

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

An example is the relation "is equal to", because if a = b is true then b = a is also true. If R^T represents the converse of R, then R is symmetric if and only if R = R^T.

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"
• "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, 2^n(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