According to the definition of a simple directed graph, the edge set can be the empty set. That is: G = [V,∅]
How would I go about proving that if a relation E is transitive, symmetric, and irreflexive, then E = ∅? I don't know where to start with this.
You can prove this by contradiction. Assume that $E \neq \emptyset$. Then there exists one edge $e \in E$. If $e = (v,v)$ for a vertex $v \in V$ then the relation is not irreflexive. Therefore $e = (v, u)$ for $v \neq u$. Since $E$ is symmetric we have that $(u,v) \in E$. We also know that $E$ is transitive and thus $(v, u), (u,v) \in E$ implies that $ (v,v) \in E$ which cannot be true since $E$ is irreflexive. Therefore, $E = \emptyset$.