The cardinality of the set `P{P[P(phi)]}` is

Options

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

Answer

To find the cardinality of the set \( P(P(P(\phi))) \), we will follow these steps:

### Step 1: Understand the notation
The notation \( P(X) \) denotes the power set of the set \( X \), which is the set of all subsets of \( X \).

### Step 2: Determine \( P(\phi) \)
The empty set \( \phi \) has only one subset, which is itself (the empty set). Therefore, the power set of the empty set is:
\[ P(\phi) = \{ \phi \} \]
This means that \( P(\phi) \) contains one element.

### Step 3: Determine \( P(P(\phi)) \)
Now we need to find the power set of \( P(\phi) \):
\[ P(P(\phi)) = P(\{ \phi \}) \]
The set \( \{ \phi \} \) has two subsets: the empty set \( \phi \) and the set \( \{ \phi \} \) itself. Therefore:
\[ P(P(\phi)) = \{ \phi, \{ \phi \} \} \]
This means that \( P(P(\phi)) \) contains two elements.

### Step 4: Determine \( P(P(P(\phi))) \)
Next, we find the power set of \( P(P(\phi)) \):
\[ P(P(P(\phi))) = P(\{ \phi, \{ \phi \} \}) \]
The set \( \{ \phi, \{ \phi \} \} \) has four subsets:
1. The empty set \( \phi \)
2. The set \( \{ \phi \} \)
3. The set \( \{ \{ \phi \} \} \)
4. The set \( \{ \phi, \{ \phi \} \} \)

Thus, we have:
\[ P(P(P(\phi))) = \{ \phi, \{ \phi \}, \{ \{ \phi \} \}, \{ \phi, \{ \phi \} \} \} \]
This means that \( P(P(P(\phi))) \) contains four elements.

### Step 5: Determine the cardinality
The cardinality of a set is the number of elements in that set. Therefore, the cardinality of \( P(P(P(\phi))) \) is:
\[ \text{Cardinality} = 4 \]

### Final Answer
The cardinality of the set \( P(P(P(\phi))) \) is \( 4 \).

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