What are all the elements in P[P{P{A}}] and how many elements are there?

[Fernando Revilla]

I quote a question from Yahoo! Answers

If A={1}.FIND NUMBER OF ELEMNTS IN P[P{P(A)}].also write all the elements?

I have given a link to the topic there so the OP can see my response.

 

[Fernando Revilla]

If $|M|$ denotes the cardinal of the set $M$ then, according to a well known property $\left|\mathcal{P}(M)\right|=2^{|M|}$. Then, $$\left|\mathcal{P}(A)\right|=2^{|A|}=2^1=2,\left|\mathcal{P}(\mathcal{P}(A))\right|=2^{ \left|\mathcal{P}(A)\right|}=2^2=4,\\\left |\mathcal{P}(\mathcal{P}(\mathcal{P}(A)))\right|=2^{ \left |\mathcal{P}(\mathcal{P}(A))\right|}=2^4=16$$
We have $\mathcal{P}(A)=\left \{\emptyset,\{1\}\right \}$ and $\mathcal{P}(\mathcal{P}(A))=\left \{\emptyset,\left \{\emptyset \right\},\left \{\{1\} \right\},\left \{\emptyset,\{1\} \right\} \right\}$. For the sake of clarity denote: $$a=\emptyset,\;b=\left \{\emptyset\right \},\;c=\left \{\{1\}\right \},\;d=\left \{\emptyset,\{1\}\right \}\qquad (*)$$
The set $\mathcal{P}(\mathcal{P}(\mathcal{P}(A)))$ is $$\mathcal{P}(\mathcal{P}(\mathcal{P}(A)))=\{ \emptyset,\left \{a\right \},\left \{b\right \},\left \{c\right \},\left \{d\right \},\left \{a,b\right \},\left \{a,c\right \},\left \{a,d\right \},\left \{b,c\right \},\left \{b,d\right \},\left \{c,d\right \},\\\left \{a,b,c\right \},\left \{a,b,d\right \},\left \{a,c,d\right \},\left \{b,c,d\right \},\left \{a,b,c,d\right \}\}$$ Now, we only need to substitute according to $(*)$. For example $\left \{b,c,d\right \}=\left \{\left \{\emptyset\right \},\left \{\{1\}\right \},\left \{\emptyset,\{1\}\right \}\right \}.$