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✗ indicates that the property may, or may not hold. All definitions tacitly require the homogeneous relation be transitive: for all if and then and there are additional properties that a homogeneous relation may satisfy. | indicates that the column's property is required by the definition of the row's term (at the very left). For example, the definition of an equivalence relation requires it to be symmetric.
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:
where the notation means that .
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]}
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 |
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^{n2−n} | 2^{n(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.