Basic Point :-
I) Power set calculates for a set but not for an element.
II) If set A have n elements, then it's powerset (i.e., P(A) ) have 2n elements.
Question :- P( P ( P ( ϕ ) ) ) ====> ϕ should be a set, let A = ϕ ====> zero elements in the set
P( ϕ ) = { ϕ } ===> 20 = 1 element in the powerset
P( P( ϕ ) ) = powerset of { ϕ } = { ϕ , { ϕ } } ===> 21 = 2 elements in the powerset.
Note that ϕ ≠ { ϕ }, those are different elements
For avoiding confusion, name them as a = ϕ , and b = { ϕ } ===> P( P( ϕ ) ) ={a,b}
P( P( P( ϕ ) ) ) = powerset of { a,b } = { ϕ, {a}, {b}, {a,b} } ===> 22 = 4 elements in the powerset.
just replace a and b, then P( P( P( ϕ ) ) ) = { ϕ, ${\color{Red} \{}$ ϕ ${\color{Red} \}}$ , ${\color{green} \{}$ ${\color{blue} \{}$ ϕ ${\color{blue} \}}$ ${\color{green} \}}$ , ${\color{orange} \{}$ ϕ , ${\color{cyan} \{}$ ϕ ${\color{cyan} \}}$ ${\color{orange} \}}$ }
for avoiding confusion, name them as p,q,r,s from left to right.
p = ϕ
q = ${\color{Red} \{}$ ϕ ${\color{Red} \}}$
r = ${\color{green} \{}$ ${\color{blue} \{}$ ϕ ${\color{blue} \}}$ ${\color{green} \}}$
s = ${\color{orange} \{}$ ϕ , ${\color{cyan} \{}$ ϕ ${\color{cyan} \}}$ ${\color{orange} \}}$
Question is What is a subset of P(P(P(ϕ))) ? ====> it means take the powerset, and check each option is belongs to the powerset or not? ( let denote A = P(P(P(ϕ))) )
Option A :- {ϕ}, it is subset of A , How? p is a element of the set ===> {p} is subset of the A
Option B :- { {ϕ} }, it is subset of A , How? q is a element of the set ===> {q} is subset of the A
Option C :- { {ϕ} , { {ϕ} } }, it is subset of A , How? q, r are the elements of the set ===> {q,r} is subset of the A
Option D :- { { { {ϕ} } } }, it is not subset of A.