In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S.[1] For example, ~ could be "is equal to".
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An irreflexive, or anti-reflexive, relation is the opposite of a reflexive relation. It is a binary relation on a set where no element is related to itself. An example is the "greater than" relation (x>y). Note that not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but others are not related to themselves (i.e. neither all nor none). For example, the binary relation "the product of x and y is even" is reflexive on the set of even numbers, irreflexive on the set of odd numbers, and neither on the set of natural numbers.
A relation is called quasi-reflexive if every element that is related to some element is related to itself. An example is the relation "has the same limit as" on the set of sequences of real numbers: Not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the same limit as some sequence, then it has the same limit as itself.
The reflexive closure of a binary relation ~ on a set S is the smallest relation ~′ such that ~′ is a superset of ~ and ~′ is reflexive on S. This is equivalent to the union of ~ and the identity relation on S. For example, the reflexive closure of x<y is x≤y.
The reflexive reduction of a binary relation ~ on a set S is the smallest relation ~′ such that ~′ shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalent to the complement of the identity relation on S with regard to ~. That is, it is equivalent to ~ except for where x~x is true. For example, the reflexive reduction of x≤y is x<y.
Examples of reflexive relations include:
Examples of irreflexive relations include:
The number of reflexive relations on an n-element set is 2n2−n.[2]
| Number of n-element binary relations of different types | ||||||||
|---|---|---|---|---|---|---|---|---|
| n | all | transitive | reflexive | preorder | partial order | total preorder | total order | equivalence relation |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 16 | 13 | 4 | 4 | 3 | 3 | 2 | 2 |
| 3 | 512 | 171 | 64 | 29 | 19 | 13 | 6 | 5 |
| 4 | 65536 | 3994 | 4096 | 355 | 219 | 75 | 24 | 15 |
| OEIS | A002416 | A006905 | A053763 | A000798 | A001035 | A000670 | A000142 | A000110 |