Three sets A, B, C are such that `A=BnnCandB=CnnA`, then

Options

A. `AsubB`
B. `AsupB`
C. `A=B`
D. `AsubB'`

Answer

To solve the problem, we need to analyze the relationships between the three sets A, B, and C given the conditions: \( A = B \cap C \) and \( B = C \cap A \).

### Step-by-Step Solution:

1. **Understanding the Given Conditions:**
We have two conditions:
- \( A = B \cap C \)
- \( B = C \cap A \)

2. **Substituting the Second Condition into the First:**
From the second condition, we know that \( B = C \cap A \). We can substitute this into the first condition:
\[ A = (C \cap A) \cap C \]
This simplifies to:
\[ A = C \cap A \]
This means that every element in A must also be in C.

3. **Analyzing the Simplified Condition:**
The equation \( A = C \cap A \) implies that all elements of A are also in C. Therefore, we can conclude:
\[ A \subseteq C \]

4. **Substituting the First Condition into the Second:**
Now, we can substitute the first condition into the second condition:
\[ B = C \cap A = C \cap (B \cap C) \]
This simplifies to:
\[ B = C \cap B \]
This means that every element in B must also be in C.

5. **Analyzing the Second Simplified Condition:**
The equation \( B = C \cap B \) implies that all elements of B are also in C. Therefore, we can conclude:
\[ B \subseteq C \]

6. **Combining the Results:**
From the above steps, we have:
- \( A \subseteq C \)
- \( B \subseteq C \)

Since both A and B are subsets of C, we can also conclude that:
\[ A \subseteq B \quad \text{and} \quad B \subseteq A \]
This leads us to the conclusion that:
\[ A = B \]

7. **Final Conclusion:**
Since \( A = B \) and both are subsets of C, we can summarize the relationships:
\[ A = B = C \]

### Final Answer:
Thus, the sets A, B, and C are equal: \( A = B = C \).