State the Axiom of Foundation and the Principle of $\in$-Induction, and show that they are equivalent in the presence of the other axioms of ZF set theory. [You may assume the existence of transitive closures.]

Given a model $(V, \in)$ for all the axioms of ZF except Foundation, show how to define a transitive class $R$ which, with the restriction of the given relation $\in$, is a model of ZF.

Given a model $(V, \in)$ of $\mathrm{ZF}$, indicate briefly how one may modify the relation $\in$ so that the resulting structure $\left(V, \in^{\prime}\right)$ fails to satisfy Foundation, but satisfies all the other axioms of $\mathrm{ZF}$. [You need not verify that all the other axioms hold in $\left(V, \in^{\prime}\right)$.]