Relation Properties via Digraphs

  1. Definition. [Anti-symmetric relations] A relation $R$ on a set $S$ is called anti-symmetric if the following is true: if the pairs $(x, y)$ and $(y, x)$ are both in $R$, then $x = y$.

    Correspondingly, if, for every two edges $(v_1, v_2)$ and $(v_2, v_1)$ that exist in the digraph $D$ of relation $R$, the values of the nodes $v_1 = v_2$ are equal, then $R$ is anti-symmetric.
    The relation corresponding to this digraph is anti-symmetric because for every two edges (v_1, v_2) and (v_2, v_1), we have v_1 = v_2.

    A digraph with edges $(v_1, v_2)$ and $(v_2, v_1)$ implying $v_1 = v_2$: anti-symmetric. Miriam Briskman, CC BY-NC 4.0.

    The relation corresponding to this digraph isn't anti-symmetric because we have at least one pair of edges (v_1, v_2) and (v_2, v_1) but v_1 != v_2.

    A digraph with at least one pair of edges $(v_1, v_2)$ and $(v_2, v_1)$ but $v_1 \neq v_2$: not anti-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.