If `A={phi,{phi}},"then P(A) is"`

Options

A. `{phi,{phi}}`
B. `{phi,{phi}},{{phi}}}`
C. `{phi,{phi},{{phi}},{phi,{phi}}}`
D. `{phi}`

Answer

To solve the problem, we need to find the power set of the set \( A = \{ \phi, \{ \phi \} \} \).

### Step-by-Step Solution:

1. **Identify the Set \( A \)**:
The set \( A \) consists of two elements: the empty set \( \phi \) and the set containing the empty set \( \{ \phi \} \). Thus, we can write:
\[ A = \{ \phi, \{ \phi \} \} \]

2. **Determine the Number of Elements in Set \( A \)**:
The set \( A \) has 2 elements. Therefore, we can denote the number of elements in \( A \) as:
\[ n(A) = 2 \]

3. **Calculate the Number of Elements in the Power Set \( P(A) \)**:
The power set \( P(A) \) of a set with \( n \) elements contains \( 2^n \) subsets. Since \( n(A) = 2 \):
\[ n(P(A)) = 2^{n(A)} = 2^2 = 4 \]

4. **List the Elements of the Power Set \( P(A) \)**:
The power set \( P(A) \) will include:
- The empty set \( \phi \)
- The set containing the first element \( \{ \phi \} \)
- The set containing the second element \( \{ \{ \phi \} \} \)
- The set containing both elements \( \{ \phi, \{ \phi \} \} \)

Therefore, we can write:
\[ P(A) = \{ \phi, \{ \phi \}, \{ \{ \phi \} \}, \{ \phi, \{ \phi \} \} \} \]

5. **Conclusion**:
The power set \( P(A) \) has 4 elements, which are:
\[ P(A) = \{ \phi, \{ \phi \}, \{ \{ \phi \} \}, \{ \phi, \{ \phi \} \} \} \]

### Final Answer:
The power set \( P(A) \) is:
\[ P(A) = \{ \phi, \{ \phi \}, \{ \{ \phi \} \}, \{ \phi, \{ \phi \} \} \} \]