Power set of a set containing a set I need some help understanding this concept a little better. I understand the general power sets, but only worked nice and easy examples where only the set consisted of only single elements like $\{a,b,c\}$. I can do that power set, but I am confused on the power set of a set containing a set. The problem I am trying to work is $\{a,\{a,b\}\}$. Typically when you have a set $\{x,y\}$ the power set is $\{\{x\},\{y\},\{x,y\}\}$ so would this problem be similar to that where $\{y\}=\{a,b\}$? Or would I then need to perform a second power set on the inner set?
 A: Actually you are not completely right when you compute the power set of $\{x,y\}$:
The power set of $\{x,y\}$ is $\{\emptyset, \{x\},\{y\},\{x,y\}\}$ i.e the empty set is always included as it is a subset of any set.
If you have the set $A = \{a, \{a,b\}\}$ and want to compute the power set, we need to find all subsets of $\{a,\{a,b\}\}$. 
The power set must have $2^2$ elements as there are 2 elements of the set.
What are these elements?
$\emptyset$ is always an element, so we include that one
$\{a\}$ is another subset as all its elements are in $A$.
Same reasoning applies to $\{\{a,b\}\}$ as $\{a,b\}$ is an element in $A$.
Finally $\{a,\{a,b\}\}$ is a subset as the entire set is always a subset.
Final powerset $\mathcal{P}(X) = \{\emptyset, \{a\}, \{\{a,b\}\},\{a,\{a,b\}\}\}$
A: Let $X=\{a,\{a,b\}\}$. The set $X$ has two elements: nothing more, nothing less. Then,
$$\mathcal P(X)=\big\{\emptyset,\{a\}, \{\{a,b\}\}, \{a,\{a,b\}\}\big\}$$
If things don't become clear, you can try "encapsulating" the element $\{a,b\}$ calling it $c$, so you can now write $X=\{a,c\}$.
