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:


 * $$\forall a \in X,\ a R a$$.


 * 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:


 * $$\forall a \in X,\ \lnot (a R a)$$.

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.

Some authors, such as Quine (1951), use the term totally reflexive for this property, and use the term relexive for the weaker property
 * $$\forall a ( \exists b (aRb \lor bRa) \to aRa).$$

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":
 * [[Image:GreaterThanOrEqualTo.png]]

Examples of irreflexive relations include:
 * "is not equal to"
 * "is coprime to"
 * "is greater than":
 * [[Image:GreaterThan.png]]

Number of reflexive relations
The formula for the number of reflexive relations is 2n 2-n