To solve this problem, we are tasked with finding the power set $P(A)$ of the given set $A$. The power set is the set of all subsets of a given set, including the empty set and the set itself.

Step 1: Understand the set

The given set $A = \{8, \$\}$. This set contains two elements: the number $8$ and the symbol $$$$$.

Step 2: Determine the subsets of $A$

For a set with $n$ elements, the total number of subsets is $2^n$. Since $A$ has $n = 2$ elements, the total number of subsets is: $2^2 = 4$

The subsets are:

1. The empty set: $\emptyset$
2. The singleton set containing $8$: $\{8\}$
3. The singleton set containing \$$$: {$}$$
4. The full set $A$: $\{8, \$\}$

Thus, the power set is: $P(A) = \{\emptyset, \{8\}, \{\$\}, \{8, \$\}\}$

Final Answer:

The power set $P(A)$ of $A = \{8, \$\}$ is: $P(A) = \{\emptyset, \{8\}, \{\$\}, \{8, \$\}\}$

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Would you like me to elaborate on any part of the solution? Here are some related questions to explore:

1. What is the general formula for the number of subsets of a set with $n$ elements?
2. How do we find the power set of a set with more than two elements?
3. Can you explain the concept of the empty set and why it is included in every power set?
4. How does the power set change if $A$ has duplicate elements, e.g., $A = \{8, 8, \$\}$?
5. How is the power set related to the concept of binary numbers?

Tip: The size of a power set grows exponentially with the number of elements in the original set!