If `A={phi,{phi}}`, then the power set P (A) of A is

Options

A. A
B. `{phi,{phi},A}`
C. `{phi,{phi},{{phi}},A}`
D. None of these

Answer

To find the power set \( P(A) \) of the set \( A = \{\emptyset, \{\emptyset\}\} \), we need to follow these steps:

### Step 1: Identify the elements of set A
The set \( A \) has two elements:
1. The empty set \( \emptyset \)
2. The set containing the empty set \( \{\emptyset\} \)

### Step 2: Determine all subsets of A
The power set \( P(A) \) consists of all possible subsets of \( A \). The subsets of a set are formed by including or excluding each element. For set \( A \), we have:

1. The empty subset: \( \emptyset \)
2. The subset containing only the first element: \( \{\emptyset\} \)
3. The subset containing only the second element: \( \{\{\emptyset\}\} \)
4. The subset containing both elements: \( \{\emptyset, \{\emptyset\}\} \)

### Step 3: Write the power set
Now, we can write the power set \( P(A) \) which includes all the subsets identified in Step 2:

\[ P(A) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\} \]

### Final Answer
Thus, the power set \( P(A) \) is:
\[ P(A) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\} \]

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