discrete math help please!?

Φ = Φ

P(Φ) = {Φ} ... (Φ has one and only one subset)

P(P(Φ)) = P({Φ}) = {Φ, {Φ}} ... (one element set has only itself and the empty set as subsets)

P(P(P(Φ))) = P({Φ, {Φ}}) = {Φ, {Φ}, {{Φ}}, {Φ, {Φ}}}

That is, the set containing:

1) The empty set

2) The set containing the first element, i.e. Φ

3) The set containing the second element, i. .{Φ}

4) The full set containing both

Hope that helps!