Relation Properties via Digraphs

  1. Definition. [Transitive relations] A relation $R$ on a set $S$ is called transitive if the following is true: if the pairs $(x, y)$ and $(y, z)$ are both in $R$, then the pair $(x, z)$ is also in $R$.

    Correspondingly, if, for every two edges $(v_1, v_2)$ and $(v_2, v_3)$ that exist in the digraph $D$ of relation $R$, the edge $(v_1, v_3)$ also exists in $D$, then $R$ is transitive.
    The relation corresponding to this digraph is transitive because for every two edges (v_1, v_2) and (v_2, v_3), there exists a third edge (v_1, v_3).

    A digraph with edges $(v_1, v_2)$ and $(v_2, v_3)$ implying the existence of $(v_1, v_3)$: transitive. Miriam Briskman, CC BY-NC 4.0.

    The relation corresponding to this digraph isn't transitive because we have at least one pair of edges (v_1, v_2) and (v_2, v_3) but there is no edge (v_1, v_3).

    A digraph with at least one pair of edges $(v_1, v_2)$ and $(v_2, v_3)$ but no edge $(v_1, v_3)$: not transitive. Miriam Briskman, CC BY-NC 4.0.

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