I would like to know the power set of $\{\{\emptyset\}\}$. The number of elements in 1 so the number of elements in the power set should be 2. Is the power set of $\{\{\emptyset\}\}$ is $\{\{\emptyset\},\{\{\emptyset\}\}\}$ ?
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The powerset of a set is the collection of all subsets of that set. So, what are the subsets of $\{\{ \emptyset \}\}$? Since the emptyset is a subset of any set, we have $\emptyset \subseteq \{\{ \emptyset \}\}$, and every set is a subset of itself, hence $\{\{\emptyset\}\} \subseteq \{\{\emptyset\}\}$. This set has no other subsets, therefore $$ \mathscr{P}( \{\{\emptyset\}\} ) = \{ \color{red}{\emptyset}, \color{green}{\{\{\emptyset\}\}}\}.$$ (I've used color here to attempt to distinguish between all of the nested parentheses. Apologies to those that are red/green colorblind—the distinction is not vital.)
answered Dec 25, 2017 at 17:34
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$\begingroup$ @KaCy: If you find this post helpful, you have the choice of marking it as the answer to your question by clicking on the checkmark on the left. $\endgroup$ Mar 27, 2018 at 7:20
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Nope, the power set of $\{\{\emptyset\}\}$ is the set of all subsets of that set, i.e. $\{\emptyset, \{\{\emptyset\}\}\}$. Maybe more helpful for you, if we call $A=\{\emptyset\}$, then the power set of $\{A\}$ is $\{\emptyset, \{A\}\}$.