The **power set** of a set \( S \) is the set of all possible subsets of \( S \), including the empty set and \( S \) itself. For the set \( S = \{ \alpha, \beta, \gamma \} \), the power set includes all combinations of the elements. 

The power set \( P(S) \) of \( S = \{\alpha, \beta, \gamma\} \) is:

\[
P(S) = \{\emptyset, \{\alpha\}, \{\beta\}, \{\gamma\}, \{\alpha, \beta\}, \{\alpha, \gamma\}, \{\beta, \gamma\}, \{\alpha, \beta, \gamma\}\}
\]

### Breakdown:
1. The empty set: \( \emptyset \)
2. Single-element subsets: \( \{\alpha\}, \{\beta\}, \{\gamma\} \)
3. Two-element subsets: \( \{\alpha, \beta\}, \{\alpha, \gamma\}, \{\beta, \gamma\} \)
4. The full set itself: \( \{\alpha, \beta, \gamma\} \)

In total, there are \( 2^3 = 8 \) subsets, which is expected since a set with \( n \) elements has \( 2^n \) subsets.

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Would you like more details or have any questions?

Here are five relative questions to explore further:
1. How do we compute the power set for larger sets?
2. What is the cardinality of a power set and how is it determined?
3. How does the concept of the power set relate to set theory in general?
4. Can a power set contain duplicate subsets?
5. How do we represent the power set of an infinite set?

**Tip:** The number of subsets in a power set is always \( 2^n \), where \( n \) is the number of elements in the original set.

Explore the power set of the set {alpha, beta, gamma} with a detailed breakdown. Learn how to compute all possible subsets, including the empty set and the full set. This problem covers set theory concepts and is suitable for students in Grades 9-12.

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