Reflexive relation

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; 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 symmetry and transitivity.

A reflexive relation that is also transitive is called a preorder.