If A = null set, then the no. of elements in `P(P(P(A)))` where `P(A)` denotes the power set of A is -

Options

A. 1
B. 2
C. 4
D. 0

Explanation

To solve the problem, we need to find the number of elements in \( P(P(P(A))) \) where \( A \) is a null set (or empty set).

### Step-by-Step Solution:

1. **Identify the Set A**:
\[ A = \emptyset \]
The null set (or empty set) has no elements.

**Hint**: Remember that the empty set is denoted by \( \emptyset \) or \( \{\} \).

2. **Find the Power Set of A**:
The power set \( P(A) \) is the set of all subsets of \( A \). Since \( A \) has no elements, the only subset of \( A \) is the empty set itself.
\[ P(A) = \{ \emptyset \} \]
Thus, \( P(A) \) contains 1 element.

**Hint**: The power set of a set with \( n \) elements has \( 2^n \) elements. For the empty set, \( n = 0 \), so \( 2^0 = 1 \).

3. **Find the Power Set of P(A)**:
Now we need to find \( P(P(A)) \). Since \( P(A) = \{ \emptyset \} \), we find the power set of this set.
The subsets of \( P(A) \) are:
- The empty set \( \emptyset \)
- The set containing the empty set \( \{ \emptyset \} \)

Therefore,
\[ P(P(A)) = \{ \emptyset, \{ \emptyset \} \} \]
Thus, \( P(P(A)) \) contains 2 elements.

**Hint**: Again, use the formula for the power set. Since \( P(A) \) has 1 element, \( P(P(A)) \) will have \( 2^1 = 2 \) elements.

4. **Find the Power Set of P(P(A))**:
Next, we need to find \( P(P(P(A))) \). Since \( P(P(A)) = \{ \emptyset, \{ \emptyset \} \} \), we find the power set of this set.
The subsets of \( P(P(A)) \) are:
- The empty set \( \emptyset \)
- The set containing the empty set \( \{ \emptyset \} \)
- The set containing the set \( \{ \emptyset \} \) \( \{ \{ \emptyset \} \} \)
- The set containing both \( \emptyset \) and \( \{ \emptyset \} \) \( \{ \emptyset, \{ \emptyset \} \} \)

Therefore,
\[ P(P(P(A))) = \{ \emptyset, \{ \emptyset \}, \{ \{ \emptyset \} \}, \{ \emptyset, \{ \emptyset \} \} \} \]
Thus, \( P(P(P(A))) \) contains 4 elements.

**Hint**: For a set with 2 elements, the power set will have \( 2^2 = 4 \) elements.

5. **Conclusion**:
The number of elements in \( P(P(P(A))) \) is 4.

**Final Answer**: 4