Let 𝐴 = {1, 2, 3}, determine all partitions of 𝐴.

A **partition** of a set is a way of dividing the set into non-empty, disjoint subsets such that every element is included in exactly one subset. For the set \( A = \{1, 2, 3\} \), we will list all possible partitions.

### 1. Partitions with one subset:
- \( \{ \{1, 2, 3\} \} \)

### 2. Partitions with two subsets:
- \( \{ \{1\}, \{2, 3\} \} \)
- \( \{ \{2\}, \{1, 3\} \} \)
- \( \{ \{3\}, \{1, 2\} \} \)

### 3. Partitions with three subsets:
- \( \{ \{1\}, \{2\}, \{3\} \} \)

In summary, the complete list of partitions of \( A = \{1, 2, 3\} \) is as follows:
1. \( \{ \{1, 2, 3\} \} \)
2. \( \{ \{1\}, \{2, 3\} \} \)
3. \( \{ \{2\}, \{1, 3\} \} \)
4. \( \{ \{3\}, \{1, 2\} \} \)
5. \( \{ \{1\}, \{2\}, \{3\} \} \)

There are a total of 5 partitions of the set \( A = \{1, 2, 3\} \).

Would you like more details or have any questions about partitions? Here are 5 related questions:

1. What is the definition of a partition of a set?
2. How do you calculate the number of partitions of a set?
3. Can you list the partitions of a set with four elements?
4. What are the Bell numbers, and how do they relate to partitions?
5. How can partitions be used in combinatorics?

**Tip:** The number of partitions of a set is given by the Bell number corresponding to the size of the set.

Discover how to determine all partitions of the set A = {1, 2, 3}. This problem explores key set theory concepts, including partitions of a set and the Bell number formula. Suitable for students in Grades 9-12, with a complete breakdown of the 5 possible partitions.

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