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Symmetric relation
Type of binary relation
Transitive binary relations
  • v
  • t
  • e
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-reflexive
Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗
Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗
Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗
Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗
Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗
Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗
Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗
Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗
Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗
Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗
Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y
Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y
Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all a , b {\displaystyle a,b} and S ≠ ∅ : {\displaystyle S\neq \varnothing :} a R b ⇒ b R a {\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}} a R b  and  b R a ⇒ a = b {\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}} a ≠ b ⇒ a R b  or  b R a {\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}} min S exists {\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}} a ∨ b exists {\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}} a ∧ b exists {\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}} a R a {\displaystyle aRa} not  a R a {\displaystyle {\text{not }}aRa} a R b ⇒ not  b R a {\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
Y indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.

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:

∀ a , b ∈ X ( a R b ⇔ b R a ) , {\displaystyle \forall a,b\in X(aRb\Leftrightarrow bRa),}

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.

Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

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Examples

In mathematics

Outside mathematics

  • "is married to" (in most legal systems)
  • "is a fully biological sibling of"
  • "is a homophone of"
  • "is a co-worker of"
  • "is a 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.

Mathematical examples
SymmetricNot symmetric
Antisymmetricequalitydivides, less than or equal to
Not antisymmetriccongruence in modular arithmetic// (integer division), most nontrivial permutations
Non-mathematical examples
SymmetricNot symmetric
Antisymmetricis the same person as, and is marriedis the plural of
Not antisymmetricis a full biological sibling ofpreys on

Properties

  • A symmetric and transitive relation is always quasireflexive.4
  • 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 n × n binary upper triangle matrices, 2n(n+1)/2.5
Number of n-element binary relations of different types
Elem­entsAnyTransitiveReflexiveSymmetricPreorderPartial orderTotal preorderTotal orderEquivalence relation
0111111111
1221211111
216134843322
3512171646429191365
465,5363,9944,0961,024355219752415
n2n22n(n−1)2n(n+1)/2∑nk=0 k!S(n, k)n!∑nk=0 S(n, k)
OEISA002416A006905A053763A006125A000798A001035A000670A000142A000110

Note that S(n, k) refers to Stirling numbers of the second kind.

Notes

See also

References

  1. Biggs, Norman L. (2002). Discrete Mathematics. Oxford University Press. p. 57. ISBN 978-0-19-871369-2. 978-0-19-871369-2 ↩

  2. "MAD3105 1.2". Florida State University Department of Mathematics. Florida State University. Retrieved 30 March 2024. https://www.math.fsu.edu/~pkirby/mad3105/index.math.htm#:~:text=%C2%A0%20Course%20Notes%3A%201.2%20Closure%20of%20Relations ↩

  3. Biggs, Norman L. (2002). Discrete Mathematics. Oxford University Press. p. 57. ISBN 978-0-19-871369-2. 978-0-19-871369-2 ↩

  4. If xRy, the yRx by symmetry, hence xRx by transitivity. The proof of xRy ⇒ yRy is similar. ↩

  5. Sloane, N. J. A. (ed.). "Sequence A006125". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩