Relation Properties via Digraphs

  1. Definition. [Symmetric relations] A relation $R$ on a set $S$ is called symmetric if, for every pair $(x, y)$ in $R$ (where $x, y \in S$,) the pair $(y, x)$ is also in $R$.

    Correspondingly, if for every edge $(v_1, v_2)$ in the digraph $D$ of relation $R$ there exists another edge in the opposite direction, $(v_2, v_1)$, then $R$ is symmetric.
    The relation corresponding to this digraph is symmetric because for every edge (v_1, v_2), we also have the edge (v_2, v_1).

    A digraph with edge $(v_1, v_2)$ implying that we also have the edge $(v_2, v_1)$: symmetric. Miriam Briskman, CC BY-NC 4.0.

    The relation corresponding to this digraph isn't symmetric because we have at least one edge of the form (v_1, v_2) without also having the edge (v_2, v_1).

    A digraph with at least one edge $(v_1, v_2)$ without the edge $(v_2, v_1)$: not symmetric. Miriam Briskman, CC BY-NC 4.0.

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