A binary relation over some set is reflexive when every element of that set is related to itself. (In symbols, a relation over a set is reflexive if .) For example, the relation 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 is neither reflexive nor anti-reflexive.
The reflexive closure of a relation is the union of with the identity relation; it is the smallest relation that is reflexive and that contains as a subset. For example, 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.