Definition:Reflexive Reduction

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 $$\mathcal{R} = \mathcal{R}^\ne$$.