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Concept
Symmetric relation
Summary
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 .
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"
"is a homophone of"
"is co-worker of"
"is 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.
A symmetric, transitive, and reflexive relation is called an equivalence relation.
Official source
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