Definition 2

A correspondence between a set A and a set B is a triple

where G is a graph such that $\mathrm{pr}_1 \mathrm{G} \subset \mathrm{A}$ and $\mathrm{pr}_2 \mathrm{G} \subset \mathrm{B}$. G is said to be the graph of $\Gamma$, A is the source, and B the target of $\Gamma$.

If $(x,\ y) \in \mathrm{G}$, we say that "$y$ corresponds to $x$ in the correspondence $\Gamma$". For each $x \in \mathrm{pr}_1 \mathrm{G}$ the correspondence $\Gamma$ is said to be defined at $x$, and $\mathrm{pr}_1 \mathrm{G}$ is called the domain of $\Gamma$; each $y \in \mathrm{pr}_2 \mathrm{G}$ is said to be a value taken by $\Gamma$, and $\mathrm{pr}_2 \mathrm{G}$ is called the range of $\Gamma$.

If $R \{ x, y \}$ is a relation which has a graph $G$ (with respect to the letters $x$, $y$), and if $A$, $B$ are two sets such that $\mathrm{pr}_1 G \subset A$ and $\mathrm{pr}_2 G \subset B$, we say that $R$ is a relation between an element of $A$ and an element of $B$ (with respect to the letters $x$, $y$). The correspondence $\Gamma = (G, A, B)$ is said to be the correspondence between $A$ and $B$ defined by the relation $R$ (with respect to $x$ and $y$).

Let G be a graph and let X be a set. The relation

implies $(x, y) \in G$ and therefore has a graph $G'$. The second projection of $G'$ consists of all the objects which correspond under G to objects of X.