The power set of {0, {0}} is O {0, (0), {{0}}, {0, {0}}} 0 {0} ○ {0, {0}} {0, {0}, {0, {0} }, { {0}, {0}}}

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### Understanding the Power Set of a Set

The concept of a power set is fundamental in the study of set theory, a branch of mathematical logic. The power set of any given set \( S \) is the set of all possible subsets of \( S \), including the empty set and \( S \) itself.

#### Question:
**What is the power set of \( \{ \emptyset, \{ \emptyset \} \} \)?**

#### Multiple Choice Options:

1. \( \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\} \} \)
2. \( \{\emptyset\} \)
3. \( \{\emptyset, \{\{\emptyset\}\}\} \)
4. \( \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}, \{\{\emptyset\}, \{\emptyset\}\} \} \)

### Explanation:
For the set \( S = \{ \emptyset, \{ \emptyset \} \} \):

- The empty set, \( \emptyset \), is always a subset of any set.
- \(\{ \emptyset \} \) is a subset.
- \(\{ \{ \emptyset \} \} \) is also a subset.
- The original set itself, \(\{ \emptyset, \{ \emptyset \} \} \), is also a subset.

Therefore, the elements that should appear in the power set are:
- The empty set: \( \emptyset \)
- The subset containing the empty set: \( \{ \emptyset \} \)
- The subset containing the singleton set of the empty set: \( \{\{ \emptyset \} \} \)
- The original set itself: \( \{ \emptyset, \{ \emptyset \} \} \)

Thus, the power set is:
\[ \{\emptyset, \{ \emptyset \}, \{\{ \emptyset \} \}, \{ \emptyset, \{ \emptyset \} \} \} \]

The correct answer is:
1. \( \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \
Transcribed Image Text:### Understanding the Power Set of a Set The concept of a power set is fundamental in the study of set theory, a branch of mathematical logic. The power set of any given set \( S \) is the set of all possible subsets of \( S \), including the empty set and \( S \) itself. #### Question: **What is the power set of \( \{ \emptyset, \{ \emptyset \} \} \)?** #### Multiple Choice Options: 1. \( \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\} \} \) 2. \( \{\emptyset\} \) 3. \( \{\emptyset, \{\{\emptyset\}\}\} \) 4. \( \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}, \{\{\emptyset\}, \{\emptyset\}\} \} \) ### Explanation: For the set \( S = \{ \emptyset, \{ \emptyset \} \} \): - The empty set, \( \emptyset \), is always a subset of any set. - \(\{ \emptyset \} \) is a subset. - \(\{ \{ \emptyset \} \} \) is also a subset. - The original set itself, \(\{ \emptyset, \{ \emptyset \} \} \), is also a subset. Therefore, the elements that should appear in the power set are: - The empty set: \( \emptyset \) - The subset containing the empty set: \( \{ \emptyset \} \) - The subset containing the singleton set of the empty set: \( \{\{ \emptyset \} \} \) - The original set itself: \( \{ \emptyset, \{ \emptyset \} \} \) Thus, the power set is: \[ \{\emptyset, \{ \emptyset \}, \{\{ \emptyset \} \}, \{ \emptyset, \{ \emptyset \} \} \} \] The correct answer is: 1. \( \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \
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