Relation Properties via Digraphs

  1. Definition. [Anti-reflexive relations] A relation $R$ on a set $S$ is called anti-reflexive if, for every $x \in S$, the pair $(x, x)$ is NOT in $R$.

    Correspondingly, if none of the vertices in the digraph $D$ of relation $R$ has a self-loop, then $R$ is anti-reflexive.
    The relation corresponding to this digraph is anti-reflexive because none of the vertices in this digraph has a self-loop.

    A digraph without any self-loops: anti-reflexive. Miriam Briskman, CC BY-NC 4.0.

    The relation corresponding to this digraph isn't anti-reflexive because at least one of the vertices in this digraph has a self-loop.

    A digraph with at least one vertex with a self-loop: not anti-reflexive. Miriam Briskman, CC BY-NC 4.0.

These notes by Miriam Briskman are licensed under CC BY-NC 4.0 and based on sources.