The power set of a set $A$, denoted as $P(A)$, is the set of all subsets of $A$, including the empty set $\emptyset$ and $A$ itself. If $A = \{1, 2, 3, 4, 5, 6\}$, let us calculate $P(A)$:

Steps:

1. The total number of subsets of $A$ is $2^n$, where $n$ is the number of elements in $A$. Here, $n = 6$, so: $2^6 = 64$ There are 64 subsets in $P(A)$.

2. List all subsets: Subsets are grouped by size, ranging from $0$ elements (the empty set) to $6$ elements (the set itself). Below is a summary:

   • Size 0: $\emptyset$ (1 subset)
   • Size 1: $\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}$ (6 subsets)
   • Size 2: $\{1, 2\}, \{1, 3\}, \dots$ (15 subsets)
   • Size 3: $\{1, 2, 3\}, \{1, 2, 4\}, \dots$ (20 subsets)
   • Size 4: $\{1, 2, 3, 4\}, \dots$ (15 subsets)
   • Size 5: $\{1, 2, 3, 4, 5\}, \dots$ (6 subsets)
   • Size 6: $\{1, 2, 3, 4, 5, 6\}$ (1 subset)

The complete list is large, but these subsets can be generated algorithmically or written explicitly if needed.

Final Answer:

The power set of $A = \{1, 2, 3, 4, 5, 6\}$, $P(A)$, consists of $64$ subsets, including the empty set $\emptyset$ and $A$ itself.

Would you like me to list the subsets explicitly or generate them in a structured table? Let me know!

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Relative Questions:

1. How does the concept of a power set relate to set operations like union and intersection?
2. What is the relationship between the power set and the binary representation of numbers?
3. How can you calculate the size of the power set for a given set?
4. Can you find $P(B)$ for $B = \{a, b, c\}$?
5. What are the practical applications of power sets in computer science and mathematics?

Tip:

When working with sets, remember that the power set grows exponentially with the size of the original set, so $n$-element sets lead to $2^n$ subsets!