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

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

### Partitions of \( A = \{1, 2, 3\} \):

1. **One subset (everything together):**
   \[
   \{\{1, 2, 3\}\}
   \]

2. **Two subsets (one element separated, two together):**
   - \(\{\{1\}, \{2, 3\}\}\)
   - \(\{\{2\}, \{1, 3\}\}\)
   - \(\{\{3\}, \{1, 2\}\}\)

3. **Three subsets (each element separated):**
   \[
   \{\{1\}, \{2\}, \{3\}\}
   \]

Thus, the complete list of partitions of \( A = \{1, 2, 3\} \) is:

\[
\begin{aligned}
1. & \ \{\{1, 2, 3\}\} \\
2. & \ \{\{1\}, \{2, 3\}\} \\
3. & \ \{\{2\}, \{1, 3\}\} \\
4. & \ \{\{3\}, \{1, 2\}\} \\
5. & \ \{\{1\}, \{2\}, \{3\}\}
\end{aligned}
\]

There are 5 partitions in total.

Would you like further details on partitions or clarification? Here are 5 related questions to deepen your understanding:

1. How many partitions would a set of 4 elements have?
2. What is the definition of a partition in terms of set theory?
3. How is the number of partitions of a set related to Bell numbers?
4. Can a partition have empty subsets?
5. What are disjoint subsets in a partition?

**Tip:** The number of partitions of a set grows rapidly with the number of elements. This sequence is known as the **Bell numbers**. For example, a set of 3 elements has 5 partitions, while a set of 4 elements has 15 partitions!

Explore how to determine all possible partitions of the set A = {1, 2, 3}. This problem introduces key concepts in set theory and combinatorics, with a detailed explanation of how the Bell numbers relate to the number of partitions. Suitable for high school students in Grades 9-12.

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