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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}.^{[2]}
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^{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.