Let P{X) denote the power set of X and A = {1,2}, then P(P(A)) contains m elements where m is equal to

Options

A. 4
B. 8
C. 16
D. 32

Answer

To solve the problem, we need to find the number of elements in the power set of the power set of the set \( A = \{1, 2\} \).

### Step-by-Step Solution:

1. **Identify the Set A**:
We have \( A = \{1, 2\} \). This set contains 2 elements.

2. **Calculate the Power Set of A**:
The power set \( P(A) \) is the set of all subsets of \( A \). The formula for the number of elements in the power set of a set with \( n \) elements is given by \( 2^n \).
- Here, \( n = 2 \) (since \( A \) has 2 elements).
- Thus, the number of elements in \( P(A) \) is \( 2^2 = 4 \).

3. **List the Elements of P(A)**:
The power set \( P(A) \) includes the following subsets:
- \( \emptyset \) (the empty set)
- \( \{1\} \)
- \( \{2\} \)
- \( \{1, 2\} \)
Therefore, \( P(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\} \).

4. **Calculate the Power Set of P(A)**:
Now, we need to find the power set of \( P(A) \), denoted as \( P(P(A)) \).
- The set \( P(A) \) has 4 elements (as calculated above).
- Using the same formula, the number of elements in \( P(P(A)) \) is \( 2^4 = 16 \).

5. **Conclusion**:
Therefore, the number of elements \( m \) in \( P(P(A)) \) is \( 16 \).

### Final Answer:
\( m = 16 \)

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