Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
reflexive
Equivalence relation Green tickY Green tickY
Preorder (Quasiorder) Green tickY
Partial order Green tickY Green tickY
Total preorder Green tickY Green tickY
Total order Green tickY Green tickY Green tickY
Prewellordering Green tickY Green tickY Green tickY
Well-quasi-ordering Green tickY Green tickY
Well-ordering Green tickY Green tickY Green tickY Green tickY
Lattice Green tickY Green tickY Green tickY Green tickY
Join-semilattice Green tickY Green tickY Green tickY
Meet-semilattice Green tickY Green tickY Green tickY
Strict partial order Green tickY Green tickY Green tickY
Strict weak order Green tickY Green tickY Green tickY
Strict total order Green tickY Green tickY Green tickY Green tickY
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all and
Green tickY indicates that the column's property is always true 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 Green tickY in the "Symmetric" column and in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation be transitive: for all if and then
A term's definition may require additional properties that are not listed in this table.

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:[1]

where the notation aRb means that (a, b) ∈ R.

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]

Examples edit

In mathematics edit

 

Outside mathematics edit

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

Relationship to asymmetric and antisymmetric relations edit

 
Symmetric 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
Symmetric Not symmetric
Antisymmetric equality divides, less than or equal to
Not antisymmetric congruence in modular arithmetic // (integer division), most nontrivial permutations
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

Properties edit

  • A symmetric and transitive relation is always quasireflexive.[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 n × n binary upper triangle matrices, 2n(n+1)/2.[3]
Number of n-element binary relations of different types
Elem­ents Any Transitive Reflexive Symmetric Preorder Partial order Total preorder Total order Equivalence relation
0 1 1 1 1 1 1 1 1 1
1 2 2 1 2 1 1 1 1 1
2 16 13 4 8 4 3 3 2 2
3 512 171 64 64 29 19 13 6 5
4 65,536 3,994 4,096 1,024 355 219 75 24 15
n 2n2 2n(n−1) 2n(n+1)/2 n
k=0
k!S(n, k)
n! n
k=0
S(n, k)
OEIS A002416 A006905 A053763 A006125 A000798 A001035 A000670 A000142 A000110

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

Notes edit

  1. ^ If xRy, the yRx by symmetry, hence xRx by transitivity. The proof of xRyyRy is similar.

References edit

  1. ^ a b Biggs, Norman L. (2002). Discrete Mathematics. Oxford University Press. p. 57. ISBN 978-0-19-871369-2.
  2. ^ "MAD3105 1.2". Florida State University Department of Mathematics. Florida State University. Retrieved 30 March 2024.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A006125". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

See also edit