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# Symmetric relation

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]

${\displaystyle \forall a,b\in X(aRb\Leftrightarrow bRa),}$

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.[citation needed]

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

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• Symmetric Relation

## Examples

### Outside mathematics

• "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

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.

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

• A symmetric and transitive relation is always quasireflexive.[a]
• A symmetric, transitive, and reflexive relation is called an equivalence relation.[1]
• 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.[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 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

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

## References

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.