1. Use the axiom of regularity to show that if x ∈ y, then y ∉ x.
2. Use the axiom of regularity to show that there cannot exist three sets w, x, and y such that w ∈ x, x ∈ y, and y ∈ w.
3. Let S = {a, b, c, d, e} and define f : S → p (S) by f (a) = {a, e}, f (b) =
{a, c, d}, f (c) = {b, d}, f (d) = ∅, and f (e) = {c, d, e}.
(a) Find the set T = {x ∈ S : x ∉ f (x)}.
(b) Note that T ∉ rng f. Is it possible to find some function g : S → p (S)0 such that T ∈ rng g, where T = {x ∈ S : x ∉ g (x)}?