Let U be the universal set and `A cup B cup C=U "Then" [ (A-B) cup (B-C) cup (C-A)]`equals

Options

A. `A cup B cup C`
B. `A cap B cap C`
C. `A cup (B cap C)`
D. `A cap (B cup C)`

Explanation

To solve the problem, we need to find the complement of the union of the sets \(A - B\), \(B - C\), and \(C - A\) given that \(A \cup B \cup C = U\).

### Step-by-Step Solution:

1. **Understanding the Sets**:
- \(A - B\) represents the elements that are in set \(A\) but not in set \(B\).
- \(B - C\) represents the elements that are in set \(B\) but not in set \(C\).
- \(C - A\) represents the elements that are in set \(C\) but not in set \(A\).

2. **Union of the Sets**:
- We need to find \( (A - B) \cup (B - C) \cup (C - A) \).
- This union will include all elements that are exclusively in \(A\), exclusively in \(B\), and exclusively in \(C\).

3. **Venn Diagram Representation**:
- To visualize this, we can draw a Venn diagram with three overlapping circles representing sets \(A\), \(B\), and \(C\).
- The regions representing \(A - B\), \(B - C\), and \(C - A\) will be the parts of the circles that do not overlap with the other circles.

4. **Finding the Complement**:
- The complement of \( (A - B) \cup (B - C) \cup (C - A) \) will include all elements that are not in these regions.
- Since \(A \cup B \cup C = U\), the complement will be the intersection of all three sets, which is \(A \cap B \cap C\).

5. **Conclusion**:
- Therefore, the final result is:
\[ [(A - B) \cup (B - C) \cup (C - A)]' = A \cap B \cap C \]

### Final Answer:
The expression \( (A - B) \cup (B - C) \cup (C - A) \) equals \( A \cap B \cap C \).