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 ''R''
T represents the
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical c ...
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.
Examples
In mathematics
* "is equal to" (
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
) (whereas "is less than" is not symmetric)
* "is
comparable to", for elements of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
* "... and ... are odd":
::::::
Outside mathematics
* "is married to" (in most legal systems)
* "is a fully biological sibling of"
* "is a
homophone
A homophone () is a word that is pronounced the same (to varying extent) as another word but differs in meaning. A ''homophone'' may also differ in spelling. The two words may be spelled the same, for example ''rose'' (flower) and ''rose'' (p ...
of"
* "is co-worker of"
* "is 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
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive.
Definition
A ho ...
is always
quasireflexive.
*A symmetric, transitive, and
reflexive relation
In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself.
An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal ...
is called an
equivalence relation.
*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 nxn binary upper triangle matrices,
References
See also
*
*
* {{annotated link, Symmetry
Binary relations