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Let's analyze each part of the question step-by-step, focusing on calculating the number of elements in the power sets of the given sets.

### a) \( \mathcal{P}(\{a, b, \{a, b\}\}) \)

The power set \( \mathcal{P}(S) \) of a set \( S \) is the set of all subsets of \( S \). The set in question here is:

\[ \{a, b, \{a, b\}\} \]

This set has three elements: \( a \), \( b \), and the set \( \{a, b\} \).

The power set of a set with \( n \) elements has \( 2^n \) elements. Since this set has 3 elements, the power set will have:

\[
2^3 = 8 \text{ elements.}
\]

### b) \( \mathcal{P}(\{\emptyset, a, \{a\}, \{\{a\}\}\}) \)

The set in question is:

\[ \{\emptyset, a, \{a\}, \{\{a\}\}\} \]

This set has four distinct elements: \( \emptyset \) (the empty set), \( a \), the set \( \{a\} \), and the set \( \{\{a\}\} \).

The power set of a set with \( n \) elements has \( 2^n \) elements. Since this set has 4 elements, the power set will have:

\[
2^4 = 16 \text{ elements.}
\]

### c) \( \mathcal{P}(\mathcal{P}(\emptyset)) \)

First, let's compute \( \mathcal{P}(\emptyset) \). The empty set \( \emptyset \) has exactly one subset, itself. So:

\[
\mathcal{P}(\emptyset) = \{\emptyset\}
\]

This set has one element, the empty set. Now, we compute the power set of \( \{\emptyset\} \), which contains 1 element. The power set of a set with 1 element contains:

\[
2^1 = 2 \text{ elements.}
\]

Thus, \( \mathcal{P}(\mathcal{P}(\emptyset)) \) has 2 elements.

### Final Answer:
- a) 8 elements
- b) 16 elements
- c) 2 elements

Would you like further details on any of the steps? Let me know!

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Here are 5 related questions you might explore:
1. What is the power set of \( \mathcal{P}(\{a, b\}) \)?
2. How many elements are in the power set of a set with 5 elements?
3. Can a power set of a set ever have fewer elements than the original set?
4. How does the power set operation behave when applied repeatedly (e.g., \( \mathcal{P}(\mathcal{P}(\mathcal{P}(S))) \))?
5. What are the elements of \( \mathcal{P}(\mathcal{P}(\{a\})) \)?

**Tip**: The number of elements in a power set grows exponentially with the number of elements in the original set!

How many elements does each of these sets have where a and b are distinct elements? 
 a) P({a, b, {a, b}}) 
 b) P({∅, a, {a}, {{a}}}) 
 c) P(P(∅))

Learn how to calculate the number of elements in the power sets of different sets where a and b are distinct elements. This problem covers set theory and the concept of power sets. Includes step-by-step calculations and explanations for a) P({a, b, {a, b}}), b) P({∅, a, {a}, {{a}}}), and c) P(P(∅)).

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