Relation Reflexivity

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Theorem

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


Proof

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

$\left({x, x}\right) \in \mathcal R \iff \neg \left({\left({x, x}\right) \notin \mathcal R}\right)$


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.

$\blacksquare$


Sources