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:
- $\tuple {x, x} \in \RR \iff \neg \paren {\tuple {x, x} \notin \RR}$
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.
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations