# Reflexive relation

(Redirected from Quasireflexive relation)

In mathematics, a binary relation R over a set X is reflexive if it relates every element of X to itself. Formally, this may be written xX : 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. However, a relation is irreflexive if, and only if, its complement is reflexive.

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, yX : x ~ y ⇒ (x ~ xy ~ 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, yX : x ~ yx ~ x and x, yX : x ~ yy ~ y, respectively. For example, a left Euclidean relation is always left, but not necessarily right, quasi-reflexive. A relation R is quasi-reflexive if, and only if, its symmetric closure RRT is left (or 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. 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 relation R is coreflexive if, and only if, its symmetric closure is anti-symmetric.

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 2n2n.

Number of n-element binary relations of different types
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 2n2 2n2n n
k=0

k! S(n, k)
n! n
k=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.