in mathematics and other areas, a binary relation r over a set x is symmetric if it holds for all a and b in x that if a is related to b then b is related to a. in mathematical notation, this is: :\forall a, b \in x,\ a r b \rightarrow \; b r a. ==examples== ===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" ==relationship to asymmetric and antisymmetric relations== by definition, a 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. {| class="wikitable" |+mathematical examples |- | || symmetric || not symmetric |- | antisymmetric || equality || "is less than or equal to" |- | not antisymmetric || congruence in modular arithmetic || "is divisible by", over the set of integers |} {| class="wikitable" |+non-mathematical examples |- | || symmetric || not symmetric |- | antisymmetric || "is the same person as, and is married" || "is the plural of" |- | not antisymmetric || "is a full biological sibling of" || "preys on" |} ==additional aspects== a symmetric relation that is also transitive and reflexive is an equivalence relation. 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. ==see also== * symmetry in mathematics * symmetry * asymmetric relation * antisymmetric relation
Mathematical_relations Symmetry