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

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The discussion focuses on calculating the number of elements in the set P[P{P(A)}] where A={1}. The cardinality of the power set is determined using the formula |P(M)|=2^|M|, leading to |P(A)|=2, |P(P(A))|=4, and |P(P(P(A)))|=16. The elements of P(A) are identified as {∅, {1}}, and P(P(A)) includes four subsets. The final set P(P(P(A))) consists of 16 distinct subsets, which are detailed through substitutions of the previously defined elements.
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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.
 
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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 $\left \{b,c,d\right \}=\left \{\left \{\emptyset\right \},\left \{\{1\}\right \},\left \{\emptyset,\{1\}\right \}\right \}.$
 
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