Find Power set of the set `A={phi,{phi}}`.

Answer

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

### Step 1: Understand the Definition of Power Set
The power set of a set \( S \) is the set of all possible subsets of \( S \), including the empty set and \( S \) itself.

**Hint:** Remember that the power set includes all combinations of elements, including no elements at all (the empty set).

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

**Hint:** List out the elements clearly to avoid confusion when forming subsets.

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

**Hint:** For each element in the set, decide whether to include it in a subset or not.

### Step 4: Compile the Power Set
Now we can compile all the subsets we found into the power set \( P(A) \):
\[ P(A) = \{\emptyset, \{\phi\}, \{\{\phi\}\}, \{\phi, \{\phi\}\}\} \]

**Hint:** Ensure that you have included all possible combinations of the elements.

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