If P(A) denotes the power set of A and A is the void set, then what is number of elements in P{P{P{P(A)}}}?

Options

A. 0
B. 1
C. 4
D. 16

Answer

To solve the problem, we need to find the number of elements in the set \( P(P(P(P(A)))) \) given that \( A \) is the empty set (void set). Let's go through the steps systematically.

### Step 1: Understand the Power Set
The power set \( P(A) \) of a set \( A \) is the set of all subsets of \( A \). If \( A \) is the empty set, then the only subset of the empty set is itself.

### Step 2: Calculate \( P(A) \)
Since \( A = \emptyset \), we have:
\[ P(A) = P(\emptyset) = \{ \emptyset \} \]
This means \( P(A) \) contains one element, which is the empty set.

### Step 3: Calculate \( P(P(A)) \)
Now we need to find \( P(P(A)) \):
\[ P(P(A)) = P(\{ \emptyset \}) \]
The set \( \{ \emptyset \} \) has two subsets: the empty set and the set itself. Therefore:
\[ P(P(A)) = \{ \emptyset, \{ \emptyset \} \} \]
This means \( P(P(A)) \) contains 2 elements.

### Step 4: Calculate \( P(P(P(A))) \)
Next, we calculate \( P(P(P(A))) \):
\[ P(P(P(A))) = P(\{ \emptyset, \{ \emptyset \} \}) \]
The set \( \{ \emptyset, \{ \emptyset \} \} \) has four subsets:
1. \( \emptyset \)
2. \( \{ \emptyset \} \)
3. \( \{ \{ \emptyset \} \} \)
4. \( \{ \emptyset, \{ \emptyset \} \} \)

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

### Step 5: Calculate \( P(P(P(P(A)))) \)
Finally, we calculate \( P(P(P(P(A)))) \):
\[ P(P(P(P(A)))) = P(\{ \emptyset, \{ \emptyset \}, \{ \{ \emptyset \} \}, \{ \emptyset, \{ \emptyset \} \} \}) \]
The set \( \{ \emptyset, \{ \emptyset \}, \{ \{ \emptyset \} \}, \{ \emptyset, \{ \emptyset \} \} \} \) has \( 4 \) elements, so it has \( 2^4 = 16 \) subsets.

### Conclusion
Thus, the number of elements in \( P(P(P(P(A)))) \) is \( 16 \).

### Final Answer
The number of elements in \( P(P(P(P(A)))) \) is **16**.

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