Question about graphs and relations

Question

If I have a directed graph $G = (V,E)$, let the relation $R$= {$(a,b)$ | $a$ has a directed path to $b$} be a relation over $V$.

How can I prove that $R$ is an equivalence relation, partial order, and total (if any are true)?

Answer 1

$R$ is clearly not in general an equivalence relation, because it’s not in general symmetric: just look at the directed graph $\bullet\longrightarrow\bullet$. On the other hand, you can’t count on its being antisymmetric, either: $\bullet\longleftrightarrow\bullet$. And it certainly needn’t be total: $\bullet\qquad\bullet$.