TITLE: Power set of a set with an empty set
QUESTION [8 upvotes]: When a set has an empty set as an element, e.g.$ \{\emptyset, a, b \}$. What is the powerset?
Is it:  $$ \{ \emptyset, \{ \emptyset \}, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$
Or
$$ \{ \emptyset, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$
Or 
$$ \{ \{\emptyset\}, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$
The confusion arises for me because, the powerset of every non-empty set has an empty set. Well the original set already has the empty set. So we don't need a subset with an empty set.
Somehow, the first one seems correct. Yet, I can't seem to accept it.

REPLY [2 votes]: Your suggestions differ by having $\emptyset$ and/or $\{\emptyset\}$ included or not.

We have $\emptyset\in\mathcal P(X)$ because $\emptyset\subseteq X$ (which would hold for any other $X$ as well)
We have $\{\emptyset\}\in\mathcal P(X)$ because $\{\emptyset\}\subseteq X$ (which is the case because $\emptyset\in X$ in this specific problem)

Therefore, your first variant is correct (and the other two are incorrect because $\emptyset\ne\{\emptyset\}$).