# Reflexive relation

In mathematics, a binary relation *R* over a set *X* is **reflexive** if it relates every element of *X* to itself.[1][2] Formally, this may be written ∀*x* ∈ *X* : *x R x*, or as I ⊆ *R* where I is the identity relation on *X*.

An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the **reflexive property** or is said to possess **reflexivity**. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.

## Related terms

A binary relation is called **irreflexive**, or **anti-reflexive**, if it doesn't relate any element to itself. An example is the "greater than" relation (*x* > *y*) on the real numbers. 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 (i.e., neither all nor none are). 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 reflexive nor irreflexive on the set of natural numbers.

A relation ~ on a set *X* is called **quasi-reflexive** if every element that is related to some element is also related to itself, formally: ∀ *x*, *y* ∈ *X* : *x* ~ *y* ⇒ (*x* ~ *x* ∧ *y* ~ *y*). 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. It does make sense to distinguish **left** and **right quasi-reflexivity**, defined by ∀ *x*, *y* ∈ *X* : *x* ~ *y* ⇒ *x* ~ *x*[3] and ∀ *x*, *y* ∈ *X* : *x* ~ *y* ⇒ *y* ~ *y*, respectively. For example, a left Euclidean relation is always left, but not necessarily right, quasi-reflexive.

A relation ~ on a set *X* is called **coreflexive** if for all *x* and *y* in *X* it holds that if *x* ~ *y* then *x* = *y*.[4] An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation. The union of a coreflexive and a transitive relation is always transitive.

A reflexive relation on a nonempty set *X* can neither be irreflexive, nor asymmetric, nor antitransitive.

The **reflexive closure** ≃ of a binary relation ~ on a set *X* is the smallest reflexive relation on *X* that is a superset of ~. Equivalently, it is the union of ~ and the identity relation on *X*, formally: (≃) = (~) ∪ (=). For example, the reflexive closure of (<) is (≤).

The **reflexive reduction**, or **irreflexive kernel**, of a binary relation ~ on a set *X* 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 *X* with regard to ~, formally: (≆) = (~) \ (=). That is, it is equivalent to ~ except for where *x*~*x* is true. For example, the reflexive reduction of (≤) is (<).

## Examples

**Examples of reflexive relations include:**

- "is equal to" (equality)
- "is a subset of" (set inclusion)
- "divides" (divisibility)
- "is greater than or equal to"
- "is less than or equal to"

**Examples of irreflexive relations include:**

- "is not equal to"
- "is coprime to" (for the integers>1, since 1 is coprime to itself)
- "is a proper subset of"
- "is greater than"
- "is less than"

## Number of reflexive relations

The number of reflexive relations on an *n*-element set is 2^{n2−n}.[5]

Elements | Any | 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 | 65,536 | 3,994 | 4,096 | 355 | 219 | 75 | 24 | 15 |

n |
2^{n2} |
2^{n2−n} |
∑nk=0 k! S(n, k) |
n! |
∑nk=0 S(n, k) | |||

OEIS | A002416 | A006905 | A053763 | A000798 | A001035 | A000670 | A000142 | A000110 |

## Philosophical logic

Authors in philosophical logic often use different terminology.
Reflexive relations in the mathematical sense are called **totally reflexive** in philosophical logic, and quasi-reflexive relations are called **reflexive**.[6][7]

## See also

- Coreflexive relation — a relation that satisfies ∀
*x*,*y*:*xRy*⇒*x*=*y* - Antisymmetric relation — a relation that satisfies ∀
*x*,*y*:*xRy*∧*yRx*⇒*x*=*y*

## Notes

- Levy 1979:74
- Relational Mathematics, 2010
- The Encyclopedia Britannica calls this property quasi-reflexivity.
- Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions to Hash Tables. In Mathematics of Program Construction (p. 337).
- On-Line Encyclopedia of Integer Sequences A053763
- Alan Hausman; Howard Kahane; Paul Tidman (2013).
*Logic and Philosophy — A Modern Introduction*. Wadsworth. ISBN 1-133-05000-X. Here: p.327-328 - D.S. Clarke; Richard Behling (1998).
*Deductive Logic — An Introduction to Evaluation Techniques and Logical Theory*. University Press of America. ISBN 0-7618-0922-8. Here: p.187

## References

- Levy, A. (1979)
*Basic Set Theory*, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover. ISBN 0-486-42079-5 - Lidl, R. and Pilz, G. (1998).
*Applied abstract algebra*, Undergraduate Texts in Mathematics, Springer-Verlag. ISBN 0-387-98290-6 - Quine, W. V. (1951).
*Mathematical Logic*, Revised Edition. Reprinted 2003, Harvard University Press. ISBN 0-674-55451-5 - Gunther Schmidt, 2010.
*Relational Mathematics*. Cambridge University Press, ISBN 978-0-521-76268-7.

## External links

- Hazewinkel, Michiel, ed. (2001) [1994], "Reflexivity",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4