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 find the power set \( P(A) \) of the set \( A = \{\phi, \{\phi\}\} \), we will follow these steps:

### Step 1: Understand the Set A
The set \( A \) contains two elements:
1. The empty set \( \phi \)
2. A set that contains the empty set \( \{\phi\} \)

So, we can write:
\[ A = \{\phi, \{\phi\}\} \]

### Step 2: Determine the Number of Subsets
The power set \( P(A) \) is the set of all possible subsets of \( A \). If a set has \( n \) elements, the power set will have \( 2^n \) subsets.

Here, \( A \) has 2 elements:
- \( \phi \)
- \( \{\phi\} \)

Thus, the number of subsets is:
\[ 2^2 = 4 \]

### Step 3: List All Subsets
Now, we will list all the subsets of \( A \):
1. The empty set: \( \emptyset \)
2. The set containing the first element: \( \{\phi\} \)
3. The set containing the second element: \( \{\{\phi\}\} \)
4. The set containing both elements: \( \{\phi, \{\phi\}\} \)

### Step 4: Write the Power Set
Now we can write the power set \( P(A) \):
\[ P(A) = \{\emptyset, \{\phi\}, \{\{\phi\}\}, \{\phi, \{\phi\}\}\} \]

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

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