Consider the following sets, where U represents a universal set:

U = {1, 2, {1}, {2}, {1, 2}} A = {1, 2, {1}} B = {{1}, {1, 2}} C = {2, {1}, {2}}.

Which one of the following is a subset of Ƥ (A)?

Select one:

a.
{1}

b.
{1, 2, {1, 2}}

c.
{{{1}}}

d.
{{{2}}}

To solve this problem, we need to determine which of the given options is a subset of the power set of \( A \), denoted as \( \mathcal{P}(A) \).

### Step 1: Determine the elements of \( A \).
Given:
\[ A = \{1, 2, \{1\}\} \]

### Step 2: Find the power set \( \mathcal{P}(A) \).
The power set \( \mathcal{P}(A) \) is the set of all subsets of \( A \). These subsets include:

1. The empty set \( \emptyset \)
2. The singleton sets \( \{1\} \), \( \{2\} \), and \( \{\{1\}\} \)
3. The pairs \( \{1, 2\} \), \( \{1, \{1\}\} \), and \( \{2, \{1\}\} \)
4. The entire set \( A = \{1, 2, \{1\}\} \)

Thus:
\[ \mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{\{1\}\}, \{1, 2\}, \{1, \{1\}\}, \{2, \{1\}\}, \{1, 2, \{1\}\}\} \]

### Step 3: Analyze the options.

Now, we check if any of the provided options is a subset of \( \mathcal{P}(A) \).

- **Option a: \( \{1\} \)**  
  \( \{1\} \) is indeed an element of \( \mathcal{P}(A) \), but \( \{1\} \) as a set is not a subset of \( \mathcal{P}(A) \), it's an element.

- **Option b: \( \{1, 2, \{1, 2\}\} \)**  
  This set contains \( 1 \), \( 2 \), and \( \{1, 2\} \). \( \{1, 2\} \) is not an element of \( \mathcal{P}(A) \), so this option cannot be a subset.

- **Option c: \( \{\{\{1\}\}\} \)**  
  This set contains \( \{\{1\}\} \), but \( \{\{1\}\} \) is not an element of \( \mathcal{P}(A) \), so this option cannot be a subset.

- **Option d: \( \{\{\{2\}\}\} \)**  
  This set contains \( \{\{2\}\} \). \( \{\{2\}\} \) is not an element of \( \mathcal{P}(A) \), so this option cannot be a subset.

### Conclusion:
None of the options presented are subsets of \( \mathcal{P}(A) \). But among the options, \( \{1\} \) (option a) **is an element** of \( \mathcal{P}(A) \). However, none of the options fit the definition of being a subset of \( \mathcal{P}(A) \) if we're strictly checking for a subset.

If I had to select the most correct answer given the choices, it would be **Option a**, but strictly speaking, this answer is based on element inclusion, not a subset.

Explore how to identify subsets in power sets with this set theory problem. Learn about the concept of power sets and subsets through detailed analysis of set elements. Suitable for high school students.

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