If `A = phi`, then find each of the following :
(i) `P(A)`
(ii) `P(P(A))`
(iii) `P{P(P(A))}`.

Answer

To solve the problem step by step, we need to find the power set of the empty set (denoted as `A = φ`) and then find the power sets of the resulting sets in each part of the question.

### Solution:

1. **Finding \( P(A) \)**:
- Given \( A = φ \) (the empty set).
- The power set \( P(A) \) is the set of all subsets of \( A \).
- The only subset of the empty set is the empty set itself.
- Therefore, \( P(A) = \{ φ \} \).

2. **Finding \( P(P(A)) \)**:
- Now we need to find the power set of \( P(A) \).
- From the previous step, we found \( P(A) = \{ φ \} \).
- The subsets of \( P(A) \) are:
- The empty set \( φ \).
- The set \( \{ φ \} \) itself.
- Therefore, \( P(P(A)) = \{ φ, \{ φ \} \} \).

3. **Finding \( P(P(P(A))) \)**:
- Now we need to find the power set of \( P(P(A)) \).
- From the previous step, we found \( P(P(A)) = \{ φ, \{ φ \} \} \).
- The subsets of \( P(P(A)) \) are:
- The empty set \( φ \).
- The set \( \{ φ \} \).
- The set \( \{ \{ φ \} \} \).
- The set \( \{ φ, \{ φ \} \} \) itself.
- Therefore, \( P(P(P(A))) = \{ φ, \{ φ \}, \{ \{ φ \} \}, \{ φ, \{ φ \} \} \} \).

### Final Answers:
- (i) \( P(A) = \{ φ \} \)
- (ii) \( P(P(A)) = \{ φ, \{ φ \} \} \)
- (iii) \( P(P(P(A))) = \{ φ, \{ φ \}, \{ \{ φ \} \}, \{ φ, \{ φ \} \} \} \)