Reflexive relation

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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:

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

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.

[edit] 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.