Let A and B be two sets such that A - B = B-A, then

Options

A. `A Delta phi = phi`
B. `A cap B = phi`
C. `A cup B = A DeltaB`
D. `A cup B = A cap B`

Answer

To solve the problem where \( A - B = B - A \), we will analyze the implications of this equation step by step.

### Step-by-Step Solution:

1. **Understanding Set Difference**:
- The expression \( A - B \) represents the elements that are in set \( A \) but not in set \( B \).
- Similarly, \( B - A \) represents the elements that are in set \( B \) but not in set \( A \).

2. **Setting Up the Equation**:
- Given \( A - B = B - A \), it means that the elements that are unique to \( A \) are the same as the elements that are unique to \( B \).

3. **Analyzing the Implications**:
- If \( A - B \) contains elements \( x \) (elements in \( A \) but not in \( B \)) and \( B - A \) contains elements \( y \) (elements in \( B \) but not in \( A \)), then we can conclude that:
- The elements in \( A \) that are not in \( B \) are equal in count and type to the elements in \( B \) that are not in \( A \).

4. **Considering the Intersection**:
- Let \( A \cap B \) be the intersection of sets \( A \) and \( B \). The elements in both sets are not counted in \( A - B \) or \( B - A \).
- Therefore, if \( A - B = B - A \), it suggests that the unique elements in both sets must balance out, leading us to consider that the intersection must also be equal.

5. **Conclusion**:
- The only way for \( A - B \) to equal \( B - A \) is if both sets \( A \) and \( B \) are equal. Thus, we conclude that:
\[ A = B \]

### Final Result:
The conclusion is that if \( A - B = B - A \), then:
\[ A = B \]