Remark 2

The notations $G(X)$ and $\Gamma(X)$ can occasionally lead to confusion with the notation introduced later (cf. no. 4, Remark following Definition 9).

Let G be a graph. Since the relation $(x, y) \in G$ implies $y \in \mathrm{pr}_2 G$, we have $G\langle X \rangle \subset \mathrm{pr}_2 G$ for every set X. Since $(x, y) \in G$ implies $x \in \mathrm{pr}_1 G$, we have $G\langle \mathrm{pr}_1 G \rangle = \mathrm{pr}_2 G$. We have $G\langle \emptyset \rangle = \emptyset$, since $x \notin \emptyset$ is a theorem. If $X \subset \mathrm{pr}_1 G$ and $X \neq \emptyset$, we have $G\langle X \rangle \neq \emptyset$.