For any three sets. `P, Q and R, S` is an element of `(P Delta Q) Delta R`, then S may belong to

Options

A. Exactly one of P,Q and R, if P, Q and R disjoint sets
B. At leasta one of P,Q and R, but not in all three of them at the same time
C. Exactly two of P, Q and R
D. Exactly one of P,Q and R or in all the three of them.

Answer

To solve the problem, we need to analyze the expression \( S \in (P \Delta Q) \Delta R \) and determine the possible membership of \( S \) in relation to the sets \( P \), \( Q \), and \( R \).

### Step-by-Step Solution:

1. **Understanding the Symmetric Difference**:
The symmetric difference between two sets \( A \) and \( B \), denoted as \( A \Delta B \), is defined as:
\[ A \Delta B = (A \setminus B) \cup (B \setminus A) \]
This means it includes elements that are in either \( A \) or \( B \) but not in both.

2. **Applying the Definition**:
We need to find \( (P \Delta Q) \Delta R \). First, we compute \( P \Delta Q \):
\[ P \Delta Q = (P \setminus Q) \cup (Q \setminus P) \]
This gives us the elements that are in \( P \) or \( Q \) but not in both.

3. **Next Step**:
Now we compute \( (P \Delta Q) \Delta R \):
\[ (P \Delta Q) \Delta R = ((P \Delta Q) \setminus R) \cup (R \setminus (P \Delta Q)) \]
Here, we are looking for elements that are in either \( (P \Delta Q) \) or \( R \) but not in both.

4. **Breaking Down the Expression**:
- **First Part**: \( (P \Delta Q) \setminus R \) includes elements that are in \( P \Delta Q \) but not in \( R \).
- **Second Part**: \( R \setminus (P \Delta Q) \) includes elements that are in \( R \) but not in \( P \Delta Q \).

5. **Combining the Results**:
Therefore, \( S \) can belong to:
- Elements that are in \( P \) or \( Q \) but not in both, and not in \( R \).
- Or elements that are in \( R \) but not in \( P \) or \( Q \).

6. **Conclusion**:
Thus, \( S \) may belong to the union of the two parts calculated above.

### Final Answer:
\( S \) may belong to the following sets:
- Elements in \( P \) or \( Q \) but not both and not in \( R \).
- Elements in \( R \) but not in \( P \Delta Q \).