If `A` is the null set, find the number of elements in the power set `P(P(A))`.

Answer

To solve the problem of finding the number of elements in the power set \( P(P(A)) \) where \( A \) is the null set, we can follow these steps:

### Step 1: Identify the null set
The null set, denoted as \( A \), is defined as:
\[ A = \emptyset \]

### Step 2: Find the power set of \( A \)
The power set \( P(A) \) is the set of all subsets of \( A \). Since \( A \) is the null set, the only subset of \( A \) is itself, which is the empty set. Therefore:
\[ P(A) = \{ \emptyset \} \]

### Step 3: Determine the number of elements in \( P(A) \)
The power set \( P(A) \) has only one element, which is the empty set:
\[ |P(A)| = 1 \]

### Step 4: Find the power set of \( P(A) \)
Now we need to find the power set of \( P(A) \), which is \( P(P(A)) \). Since \( P(A) \) contains one element (the empty set), we can find its power set:
\[ P(P(A)) = P(\{ \emptyset \}) \]
The power set of a set with one element contains two subsets: the empty set and the set itself. Therefore:
\[ P(P(A)) = \{ \emptyset, \{ \emptyset \} \} \]

### Step 5: Determine the number of elements in \( P(P(A)) \)
The power set \( P(P(A)) \) has two elements:
\[ |P(P(A))| = 2 \]

### Final Answer
Thus, the number of elements in the power set \( P(P(A)) \) is:
\[ \boxed{2} \]