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Set(Problem Related to Subset)

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Which of the following is not a subset of $P(P(P(\phi)))$?

$A)$ {$\phi$}

$B)$ { {$\phi$} }

$C)$  { {$\phi$} ,  { {$\phi$} } }

$D)$  { { { {$\phi$} } } }
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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.

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