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A symmetric relation is a type of
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
. 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: :\forall a, b \in X(a R b \Leftrightarrow b R a) , where the notation aRb means that (a,b)\in R. If ''R''T represents the
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a categorical or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities 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 upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not. In mathem ...
* "... 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 pronouncedPronunciation is the way in which a word or a language is spoken. This may refer to generally agreed-upon sequences of sounds used in speaking a given word or language in a specific dialect ("correct ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is always quasireflexive. *A symmetric, transitive, and
reflexive relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is called an
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. *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, 2^.


References


See also

* * * {{annotated link, Symmetry Binary relations