# Relation Reflexivity

## Theorem

Every relation has exactly one of these properties: it is either:

reflexive,
antireflexive or
non-reflexive.

## Proof

A reflexive relation can not be antireflexive, and vice versa:

$\tuple {x, x} \in \RR \iff \neg \paren {\tuple {x, x} \notin \RR}$

By the definition of non-reflexive, a reflexive relation can not also be non-reflexive.

So a reflexive relation is neither antireflexive nor non-reflexive.

An antireflexive relation can be neither reflexive (see above) nor non-reflexive (by the definition of non-reflexive).

By its own definition, if a relation is neither reflexive nor antireflexive, then it is non-reflexive.

$\blacksquare$