Power Set of set containing the empty set and a set as an element I'm just a little confused on the nuances of power sets. I'm looking for the power set of the following set: $\{\varnothing, a, \{a\}\}$
 A: It's 
$\mathfrak{P}(M)=\{ \emptyset, \{\emptyset\}, \{a\}, \{\{a\}\}, \{\emptyset, a\}, \{\emptyset, \{a\}\}, \{a, \{a\}\},  \{\emptyset, a,\{a\}\}    \}$ where $M=\{\emptyset, a,\{a\}\}$.
A: Here's how I solve such problems. 
First off, keep in mind that for a set $S$ containing $n$ distinct elements, where $n$ is a nonnegative integer, $|\wp(S)| = 2^n $.
For your set $M = \{\emptyset, a,\{a\}\}$, M contains 3 distinct elements, therefore it should have $2^3 = 8$ distinct subsets.
Ok, now when you are listing the subsets of a set, start off by writing down the empty set $\emptyset$, and the set itself $\{\emptyset, a,\{a\}\}$.
Next, write out each element in the set as a subset of its own, that is $\{\emptyset\}, \{a\}$ and  $\{\{a\}\}$. That takes care of 5 subsets so we have 3 left.
Finally, start making different pairs of each elements, that is $\{\emptyset, a\}, \{\emptyset, \{a\}\}$ and $\{a, \{a\}\}$.
Putting it all together:
$\wp(M)= \{\emptyset, \{\emptyset\}, \{a\}, \{\{a\}\}, \{\emptyset, a\}, \{\emptyset, \{a\}\}, \{a, \{a\}\},  \{\emptyset, a,\{a\}\}\}$.
