Let `A, B, C` be three events such that A and B are independent and `P(C) = 0`, then events `A, B ,C` are

Options

A. A and C are independent
B. B and C are independent
C. A, B and C are independent
D. all of these

Answer

To solve the problem, we need to analyze the relationships between the events A, B, and C based on the information provided.

### Step-by-step Solution:

1. **Understanding Independence**:
- Two events A and B are said to be independent if \( P(A \cap B) = P(A) \cdot P(B) \).
- We are given that A and B are independent.

2. **Probability of Event C**:
- We are given that \( P(C) = 0 \). This means that event C cannot occur; there are no favorable outcomes for event C.

3. **Independence of C with A and B**:
- To check if event C is independent of A, we need to see if \( P(A \cap C) = P(A) \cdot P(C) \).
- Since \( P(C) = 0 \), we have \( P(A \cap C) = P(A) \cdot 0 = 0 \).
- The probability \( P(A \cap C) \) is also 0 because C cannot occur. Thus, A and C are independent.

4. **Independence of C with B**:
- Similarly, for event B, we check if \( P(B \cap C) = P(B) \cdot P(C) \).
- Again, since \( P(C) = 0 \), we have \( P(B \cap C) = P(B) \cdot 0 = 0 \).
- The probability \( P(B \cap C) \) is also 0, which means B and C are independent.

5. **Conclusion about Independence**:
- Since A and B are independent, A and C are independent, and B and C are independent, we can conclude that all three events A, B, and C are independent.

### Final Answer:
Thus, the correct option is **D: all of these**. Events A, B, and C are independent.