Relation Properties via Digraphs

Since, as we learned, relations can be depicted using diagraphs (or graphs,) we can find what properties (reflexive, anti-reflexive, symmetric, anti-symmetric, and transitive) a relation has by looking at its digraph.

  1. Definition. [Reflexive relations] A relation $R$ on a set $S$ is called reflexive if, for every $x \in S$, the pair $(x, x)$ is in $R$.

    Correspondingly, if every vertex in the digraph $D$ of relation $R$ has a self-loop, then $R$ is reflexive.
    The relation corresponding to this digraph is reflexive because each of the vertices in this digraph has a self-loop.

    A digraph with vertices each having a self-loop: reflexive. Miriam Briskman, CC BY-NC 4.0.

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

    A digraph with at least one vertex without a self-loop: not 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.