To solve the problem, we need to find $\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset)))))$, where $\mathcal{P}(S)$ denotes the power set of the set $S$. Let's go through the steps carefully:

Step 1: Start with $\emptyset$ (the empty set)

The empty set is denoted as: $\emptyset = \{\}$

Step 2: Find $\mathcal{P}(\emptyset)$

The power set of a set is the set of all its subsets. The only subset of $\emptyset$ is $\emptyset$ itself. Thus, the power set of $\emptyset$ is: $\mathcal{P}(\emptyset) = \{\emptyset\}$

Step 3: Find $\mathcal{P}(\mathcal{P}(\emptyset))$

Now, we need to find the power set of $\mathcal{P}(\emptyset) = \{\emptyset\}$. The subsets of $\{\emptyset\}$ are:

• $\emptyset$ (the empty subset)
• $\{\emptyset\}$ (the set itself)

So, the power set is: $\mathcal{P}(\mathcal{P}(\emptyset)) = \{\emptyset, \{\emptyset\}\}$

Step 4: Find $\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset)))$

Now, we find the power set of $\mathcal{P}(\mathcal{P}(\emptyset)) = \{\emptyset, \{\emptyset\}\}$. The subsets of this set are:

• $\emptyset$ (the empty subset)
• $\{\emptyset\}$
• $\{\{\emptyset\}\}$
• $\{\emptyset, \{\emptyset\}\}$

So, the power set is: $\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\}$

Step 5: Find $\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))))$

Finally, we find the power set of $\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\}$. The subsets of this set are:

• $\emptyset$
• $\{\emptyset\}$
• $\{\{\emptyset\}\}$
• $\{\emptyset, \{\emptyset\}\}$
• $\{\{\{\emptyset\}\}\}$
• $\{\emptyset, \{\{\emptyset\}\}\}$
• $\{\{\emptyset\}, \{\{\emptyset\}\}\}$
• $\{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}$

Thus, the final power set is: $\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset)))) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}, \{\{\{\emptyset\}\}\}, \{\emptyset, \{\{\emptyset\}\}\}, \{\{\emptyset\}, \{\{\emptyset\}\}\}, \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}\}$

Would you like more details or clarification on any of these steps?

Here are 5 related questions to expand on this:

1. How is the power set defined for any set?
2. What are the elements of the power set of a set with 3 elements?
3. How many elements does the power set of a set with $n$ elements have?
4. What is the relationship between the cardinality of a set and the cardinality of its power set?
5. Can you find the power set of a non-empty set, say $\{1, 2\}$?

Tip: The cardinality of the power set $\mathcal{P}(S)$ of a set $S$ with $n$ elements is always $2^n$, since every element of $S$ can either be included or not included in a subset.