All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive: for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table.
A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if:[1]
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 RT represents the converse of R, then R is symmetric if and only if R = RT.[2]
Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.[1]
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.
Note that S(n, k) refers to Stirling numbers of the second kind.
Italian politician (born 1938) Giuliano AmatoOMRI GCMGGiuliano Amato in 2013Prime Minister of ItalyIn office26 April 2000 – 11 June 2001PresidentCarlo Azeglio CiampiPreceded byMassimo D'AlemaSucceeded bySilvio BerlusconiIn office28 June 1992 – 29 April 1993PresidentOscar Luigi ScalfaroPreceded byGiulio AndreottiSucceeded byCarlo Azeglio CiampiDeputy Prime Minister of ItalyIn office29 July 1987 – 13 April 1988Prime MinisterGiovanni GoriaPreceded byArnaldo ForlaniS…
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