Power set of $\{\varnothing, \{ \varnothing\}\}$ The power set of $\{\varnothing, \{ \varnothing\}\}$ is defined as follows:
$$\{\varnothing, \{ \varnothing\}, \{\{\varnothing\}\}, \{\varnothing, \{\varnothing\}\}\}$$
Why do we have the 3rd element in the answer $\{\{\varnothing\}\}$?
 A: If we have a set with two elements $X=\{a,b\}$, then the powerset of $X$ is: $\{∅, \{a\},\{b\},\{a,b\}\}$, just sub in your elements as $a$ and $b$.
I think the fact that the elements in the set are the empty set or the set containing the empty set is what is confusing you.
For more general practice consider $ Y = \emptyset $, then $\mathcal{P}(Y) = \{ \emptyset\}$
Further, let $Z = \{  \emptyset \}$, then  $\mathcal{P}(Z) = \{ \emptyset, \{ \emptyset \}\}$
A: Power set is the set of all subsets. Each element is a subset of the initial set.
The initial set is $\{\color{red}\emptyset, \color{blue}{\{\emptyset\}}\}$
That is a set with two elements $\color{red}\emptyset$ and $\color{blue}{\{\emptyset\}}$
So what are the subsets of a set with two elements.  There are $2^2 =4$ such sets because each element can either be in as set or not.  So the four subsets are:

*

*The set with neither $\color{red}\emptyset$ nor $\color{blue}{\{\emptyset\}}$.
That is the set: $\emptyset$

*The set that has  $\color{red}\emptyset$ but not $\color{blue}{\{\emptyset\}}$.
That is the set: $\{\color{red}\emptyset\}$.

*The set that does not have $\color{red}\emptyset$ but has $\color{blue}{\{\emptyset\}}$
That is the set: $\{\color{blue}{\{\emptyset\}}\}$

*The set that has both $\color{red}\emptyset$ and $\color{blue}{\{\emptyset\}}$.
That is the set $\{\color{red}\emptyset,\color{blue}{\{\emptyset\}}\}$
