Definition:Reflexive Reduction

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

The reflexive reduction of $\RR$ is denoted $\RR^\ne$, and is defined as:


 * $\RR^\ne := \RR \setminus \set {\tuple {x, x}: x \in S}$

Also see

 * Reflexive Reduction is Largest Antireflexive Relation which is Subset

From Set Difference as Intersection with Relative Complement and Intersection is Largest Subset, it follows that $\RR^\ne$ is the largest antireflexive relation on $S$ which is contained in $\RR$.


 * Antireflexive Relation equals its Reflexive Reduction

Thus if $\RR$ is antireflexive, then $\RR = \RR^\ne$.