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To find the power set of the given sets, we need to first understand the elements within each set and then determine all possible subsets.

### (f) \(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset)))\)
1. **Step 1: Compute \(\mathcal{P}(\emptyset)\):**
   - The empty set \(\emptyset\) has only one subset, which is itself.
   - \(\mathcal{P}(\emptyset) = \{\emptyset\}\).

2. **Step 2: Compute \(\mathcal{P}(\mathcal{P}(\emptyset))\):**
   - The set \(\mathcal{P}(\emptyset)\) has one element, \(\emptyset\).
   - The subsets of \(\{\emptyset\}\) are \(\emptyset\) and \(\{\emptyset\}\).
   - \(\mathcal{P}(\mathcal{P}(\emptyset)) = \{\emptyset, \{\emptyset\}\}\).

3. **Step 3: Compute \(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset)))\):**
   - The set \(\mathcal{P}(\mathcal{P}(\emptyset))\) has two elements: \(\emptyset\) and \(\{\emptyset\}\).
   - The subsets of \(\{\emptyset, \{\emptyset\}\}\) are: \(\emptyset\), \(\{\emptyset\}\), \(\{\{\emptyset\}\}\), and \(\{\emptyset, \{\emptyset\}\}\).
   - \(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\}\).

### (g) \(\mathcal{P}(\{{a}, \{a, b\}\})\)
1. **Identify Elements of the Set:**
   - The set is \(\{{a}, \{a, b\}\}\).
   - It has two elements: \(a\) and \(\{a, b\}\).

2. **Find All Subsets:**
   - The subsets are: 
     - \(\emptyset\) (the empty set),
     - \(\{a\}\) (subset containing the element \(a\)),
     - \(\{\{a, b\}\}\) (subset containing the element \(\{a, b\}\)),
     - \(\{a, \{a, b\}\}\) (subset containing both elements).

3. **Power Set:**
   - \(\mathcal{P}(\{{a}, \{a, b\}\}) = \{\emptyset, \{a\}, \{\{a, b\}\}, \{a, \{a, b\}\}\}\).

These are the power sets for each of the given sets. 

Would you like further details or explanations on any of the steps? Here are 5 related questions you might be interested in:

1. What is the general formula for the number of elements in the power set of a given set?
2. How does the structure of a set influence the structure of its power set?
3. What are the practical applications of power sets in computer science?
4. How do power sets relate to boolean algebras?
5. How would you compute the power set of a set with three or more elements?

**Tip:** For any set with \( n \) elements, its power set will contain \( 2^n \) elements.

Find the power set of the following sets: 

f. \(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset)))\)
g. \(\mathcal{P}(\{{a}, \{a, b\}\})\)

Discover how to find the power sets of complex nested sets and multi-element sets using step-by-step solutions. This problem explores the power set of \(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset)))\) and \(\mathcal{P}(\{{a}, \{a, b\}\})\), covering key set theory concepts.

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