= Solution
A graph must assign exactly one output to every input, while the distinguished unary predicate must select exactly one element. Suitable sentences are
$$
\boxed{\phi=\forall x\exists y\,[P(x,y)\land\forall z(P(x,z)\to z=y)],}
$$
$$
\boxed{\psi=\exists x\,[Q(x)\land\forall z(Q(z)\to z=x)].}
$$
The existential clauses provide totality and nonemptiness of the selected output, respectively; the universal clauses provide uniqueness. Thus a structure satisfying $\phi,\psi$ uniquely determines a total unary function $f$ and a constant $c$ on its unchanged domain. Conversely any interpretations of $f,c$ determine such relations by $P(x,y)\iff f(x)=y$ and $Q(x)\iff x=c$.
Back to article page
