Transitive binary relations  

 
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 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 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 R^{T} represents the converse of R, then R is symmetric if and only if R = R^{T}.^{[citation needed]}
Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.^{[1]}
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Reflexive, Symmetric, and Transitive Relations on a Set

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L2.5: Symmetric Relation with examples  Discrete Maths

Reflexive, Symmetric, Transitive Tutorial

Symmetric Relation
Transcription
Examples
In mathematics
 "is equal to" (equality) (whereas "is less than" is not symmetric)
 "is comparable to", for elements of a partially ordered set
 "... and ... are odd":
Outside mathematics
 "is married to" (in most legal systems)
 "is a fully biological sibling of"
 "is a homophone of"
 "is coworker 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, 2^{n(n+1)/2}.^{[2]}
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  2^{n2}  2^{n(n−1)}  2^{n(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
 ^ If xRy, the yRx by symmetry, hence xRx by transitivity. The proof of xRy ⇒ yRy is similar.
References
 ^ ^{a} ^{b} ^{c} Biggs, Norman L. (2002). Discrete Mathematics. Oxford University Press. p. 57. ISBN 9780198713692.
 ^ Sloane, N. J. A. (ed.). "Sequence A006125". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation.
See also
 Commutative property – Property of some mathematical operations
 Symmetry in mathematics
 Symmetry – Mathematical invariance under transformations