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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.

What is the value of P(P(∅)), where P is the power set and ∅ is the empty set?

Discover how to compute P(P(∅)), the power set of the power set of the empty set. This problem uses core set theory concepts such as subsets and power sets, and includes step-by-step reasoning. Ideal for Grades 9-12.

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