Reflexive relation

In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity.

At least in this context, (binary) relation (on X) always means a relation on X×X, or in other words from a set X into itself.

• A reflexive relation R on set X is one where for all a in X, a is R-related to itself. In mathematical notation, this is:

$\forall a\in X,\ aRa$

• An irreflexive (or aliorelative) relation R is one where for all a in X, a is never R-related to itself. In mathematical notation, this is:

$\forall a\in X,\ \lnot (aRa)$.

The reflexive closure R ⁼ is defined as R ⁼ = {(x, x) | x ∈ X} ∪ R, i.e., the smallest reflexive relation over X containing R. This can be seen to be equal to the intersection of all reflexive relations containing R.

The reflexive reduction of a binary relation R on a set is the irreflexive relation R' with xR'y iff xRy for all x≠y.

Note: A common misconception is that a relationship is always either reflexive or irreflexive. Irreflexivity is a stronger condition than failure of reflexivity, so a binary relation may be reflexive, irreflexive, or neither. The strict inequalities "less than" and "greater than" are irreflexive relations whereas the inequalities "less than or equal to" and "greater than or equal to" are reflexive. However, if we define a relation R on the integers such that a R b iff a = -b, then it is neither reflexive nor irreflexive, because 0 is related to itself.

A transitive irreflexive relation is an asymmetric relation and a strict partial order, while a transitive reflexive relation is only a preorder. Thus on a finite set there are more of the latter than of the former.

Properties containing the reflexive property

Preorder - A reflexive relation that is also transitive. Special cases of preorders such as partial orders and equivalence relations are, therefore, also reflexive.

Examples

Examples of reflexive relations include:

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

Examples of irreflexive relations include:

• "is not equal to"
• "is coprime to"
• "is greater than":

Number of reflexive relations

Number of n-element binary relations of different types

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	2n2		2n(n−1)	2n(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.