Reflexive relation

by Ryan Hendrickson Jul 15 2016

A binary relation over some set is reflexive when every element of that set is related to itself. (In symbols, a relation $~$R$~$ over a set $~$X$~$ is reflexive if $~$\forall a \in X, aRa$~$.) For example, the relation $~$\leq$~$ defined over the real numbers is reflexive, because every number is less than or equal to itself.

A relation is anti-reflexive when no element of the set over which it is defined is related to itself. $~$<$~$ is an anti-reflexive relation over the real numbers. Note that a relation doesn't have to be either reflexive or anti-reflexive; if Alice likes herself but Bob doesn't like himself, then the relation "_ likes _" over the set $~$\{Alice, Bob\}$~$ is neither reflexive nor anti-reflexive.

The reflexive closure of a relation $~$R$~$ is the union of $~$R$~$ with the [Identity_relation identity relation]; it is the smallest relation that is reflexive and that contains $~$R$~$ as a subset. For example, $~$\leq$~$ is the reflexive closure of $~$<$~$.

Some other simple properties that can be possessed by binary relations are [Symmetric_relation symmetry] and transitivity.

A reflexive relation that is also transitive is called a [Preorder preorder].