Reflexive Reduction is Antireflexive

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\RR$ be a relation on a set $S$.

Let $\RR^\ne$ denote the reflexive reduction of $\RR$.


Then $\RR^\ne$ is antireflexive.


Proof

By the definition of reflexive reduction:

$\RR^\ne = \RR \setminus \Delta_S$

where $\Delta_S$ denotes the diagonal relation on $S$.

By Set Difference Intersection with Second Set is Empty Set:

$\paren {\RR \setminus \Delta_S} \cap \Delta_S = \O$

Hence by Relation is Antireflexive iff Disjoint from Diagonal Relation, $\RR^\ne$ is antireflexive.

$\blacksquare$


Sources