The expression provided is $P(P(\emptyset))$, where $P$ denotes the power set and $\emptyset$ represents the empty set.

Let's solve it step by step:

1. Find $P(\emptyset)$:

   • The power set $P(S)$ of a set $S$ is the set of all subsets of $S$.
   • The empty set, $\emptyset$, has only one subset: itself. Thus, the power set of the empty set is $P(\emptyset) = \{ \emptyset \}$.

2. Find $P(P(\emptyset))$:

   • Now we need to find the power set of $\{ \emptyset \}$.
   • The set $\{ \emptyset \}$ has two subsets: the empty set $\emptyset$ and the set itself $\{ \emptyset \}$.
   • Therefore, $P(\{ \emptyset \}) = \{ \emptyset, \{ \emptyset \} \}$.

Hence, the final result is: $P(P(\emptyset)) = \{ \emptyset, \{ \emptyset \} \}.$

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What is the power set of a set with three elements?
2. How many elements does the power set of a set with $n$ elements contain?
3. What is the relationship between a set and its power set in terms of cardinality?
4. How does the concept of power sets apply to infinite sets?
5. Can the power set of a set contain elements other than sets?

Tip: The power set of a set with $n$ elements always contains $2^n$ elements.