Power set of a set with an empty set 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.
 A: The first one is correct.
This is because $\emptyset$ and $\{\emptyset\}$ are different. The first is an empty set whereas the second is a set whose only element is the empty set.
Both are subsets of the given set. This is because the $\emptyset$ is the subset of every set, and as it happens to be an element of the given set, the set containing it as its element is also its subset.
A: If a set $A$ is such that $\emptyset\in A$, its power set must necessarily contain these two sets:


*

*$\emptyset$ (like all other power sets), corresponding to selecting nothing from $A$ (not even $\emptyset$, which is something)

*$\{\emptyset\}$, corresponding to selecting $\emptyset$ only


Therefore only the first of your proposed answers is correct, as you think.
A: 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 $\emptyset\ne\{\emptyset\}$).
