Symmetric relation

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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 means that .

If RT represents the converse of R, then R is symmetric if and only if R = RT.[citation needed]

Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.[1]

Examples[edit]

In mathematics[edit]

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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]

  • 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]
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 2n2n 2n(n+1)/2 n!
OEIS A002416 A006905 A053763 A006125 A000798 A001035 A000670 A000142 A000110

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

References[edit]

  1. ^ a b c Biggs, Norman L. (2002). Discrete Mathematics. Oxford University Press. p. 57. ISBN 978-0-19-871369-2.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A006125". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

See also[edit]