State the Axiom of Foundation and the Principle of $\epsilon$-Induction, and show that they are equivalent (in the presence of the other axioms of $Z F$ ). [You may assume the existence of transitive closures.]

Explain briefly how the Principle of $\epsilon$-Induction implies that every set is a member of some $V_{\alpha}$.

Find the ranks of the following sets:

(i) $\{\omega+1, \omega+2, \omega+3\}$,

(ii) the Cartesian product $\omega \times \omega$,

(iii) the set of all functions from $\omega$ to $\omega^{2}$.

[You may assume standard properties of rank.]