If `A={1,B}`,where `B = {2,3}`.Then, `P(A)` is equal to

Options

A. `{{1}, {B}, {1, B}}`
B. `{ phi, { 1} , { B } , { 1, B } }`
C. `{ phi, { 1 } , { B } }`
D. None of these

Answer

To find the power set \( P(A) \) of the set \( A = \{1, B\} \) where \( B = \{2, 3\} \), we will follow these steps:

### Step 1: Understand the Set
First, we need to clarify what the set \( A \) looks like. Given that \( B = \{2, 3\} \), we can rewrite set \( A \):
\[ A = \{1, \{2, 3\}\} \]

### Step 2: Determine the Subsets of \( A \)
Next, we need to find all the subsets of \( A \). The subsets of a set can be found by considering all combinations of its elements. For a set with \( n \) elements, there are \( 2^n \) subsets.

Since \( A \) has 2 elements (1 and \( B \)), the number of subsets will be:
\[ 2^2 = 4 \]

### Step 3: List All Subsets
The subsets of \( A \) are:
1. The empty set: \( \emptyset \)
2. The set containing the first element: \( \{1\} \)
3. The set containing the second element (which is \( B \)): \( \{\{2, 3\}\} \)
4. The set containing both elements: \( \{1, \{2, 3\}\} \)

Thus, the subsets of \( A \) are:
\[ \emptyset, \{1\}, \{\{2, 3\}\}, \{1, \{2, 3\}\} \]

### Step 4: Form the Power Set
The power set \( P(A) \) is the set of all subsets of \( A \). Therefore, we can write:
\[ P(A) = \{\emptyset, \{1\}, \{\{2, 3\}\}, \{1, \{2, 3\}\}\} \]

### Final Answer
The power set \( P(A) \) is:
\[ P(A) = \{\emptyset, \{1\}, \{B\}, A\} \]
Where \( B = \{2, 3\} \).