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:


 * $$\mathcal R^\ne \ \stackrel {\mathbf {def}} {=\!=} \ \mathcal R - \left\{{\left({x, x}\right): x \in S}\right\}$$

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

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