Definition:Reflexive Reduction

Definition
Let $\mathcal R$ be a relation on a set $S$.

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


 * $\mathcal R^\ne := \mathcal R \setminus \left\{{\left({x, x}\right): x \in S}\right\}$

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 $\mathcal R^\ne$ is the largest antireflexive relation on $S$ which is contained in $\mathcal R$.


 * Antireflexive Relation equals its Reflexive Reduction

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