Definition 1

G is said to be a graph if every element of G is an ordered pair, i.e., if the relation

is true.

If G is a graph, the relation $(x, y) \in G$ is expressed by saying that "$y$ corresponds to $x$ under G".

Let $\boldsymbol{R}\{x, y\}$ be a relation, where $x$ and $y$ are distinct letters. Let $\boldsymbol{G}$ be a letter, distinct from $x$ and $y$, which does not appear in $\boldsymbol{R}$. If the relation

is true, the relation $\boldsymbol{R}$ is said to have a graph (with respect to the letters $x$ and $y$). The graph $\boldsymbol{G}$ is then unique by virtue of the axiom of extent, and is called the graph of $\boldsymbol{R}$ with respect to $x$ and $y$.

¶ Let $Z$ be a letter, distinct from $x$ and $y$, which does not appear in $R$. If the relation

is true, then $R$ has a graph; we may take this graph to be the set of ordered pairs $z$ such that $z \in Z$ and $R \{ \mathrm{pr}_1 z,\ \mathrm{pr}_2 z \}$ ($z$ being a letter distinct from $x$, $y$, $Z$ which does not appear in $R$). This condition is satisfied if we know a term $T$, which does not contain either $x$ or $y$, such that $R \Rightarrow ((x,\ y) \in T)$ is true.