Proposition 1

Let G be a graph. There exists exactly one set A and exactly one set B with the following properties :

(1) the relation $(\exists y)((x,\ y) \in \mathrm{G})$ is equivalent to $x \in \mathrm{A}$;

(2) the relation $(\exists x)((x,\ y) \in \mathrm{G})$ is equivalent to $y \in \mathrm{B}$.

For it is sufficient to take A (resp. B) to be the set of all objects of the form $\mathrm{pr}_1 z$ (resp. $\mathrm{pr}_2 z$), where $z \in \mathrm{G}$ (§ 1, no. 6). Precisely,

these sets are called respectively the first and second projections of the graph G, or the domain and range of G; they are denoted by $\mathrm{pr}_1 \langle \mathrm{G} \rangle$ and $\mathrm{pr}_2 \langle \mathrm{G} \rangle$ (or by $\mathrm{pr}_1 \mathrm{G}$ and $\mathrm{pr}_2 \mathrm{G}$ if there is no risk of confusion). It is immediately verified that $\mathrm{G} \subset (\mathrm{pr}_1 \mathrm{G}) \times (\mathrm{pr}_2 \mathrm{G})$; every set of ordered pairs is therefore a subset of a product, and conversely. If one of the two sets $\mathrm{pr}_1 \mathrm{G}$, $\mathrm{pr}_2 \mathrm{G}$ is empty, we have $\mathrm{G} = \emptyset$ (§ 2, Proposition 2).