Past exam of the logic and set theory course of the University of Oxford / 2011 / A / 2 / d

The antecedent says that the nonempty domain has at most two elements: its two witnesses need not be distinct. The consequent says it has exactly one element. ThusOne-element structures satisfy both sides; domains of size at least three, including infinite domains, make the antecedent false. The function $f$ has no bearing on this condition.

A two-element model of $\varphi$ would be a finite model with even size, contrary to the defining property established in part (b). Every model of $\varphi$ therefore has size other than two, and the preceding interpretation of $\psi$ gives $\boxed{\{\varphi \}\models \psi }$. This includes infinite models, since the antecedent of $\psi$ is false there.