Which one of the following is `(A-B)uu(B-A)?`

Options

A. `(AuuB)uu(A-B)`
B. `(AuuB)uu(AnnB)`
C. `(AuuB)-(AnnB)`
D. `(A-B)nn(B-A)`

Answer

To solve the question, we need to find the expression for \( (A - B) \cup (B - A) \). Let's break it down step by step.

### Step 1: Understand the operations involved
- **Set Difference**: \( A - B \) means the elements that are in set \( A \) but not in set \( B \).
- **Set Difference**: \( B - A \) means the elements that are in set \( B \) but not in set \( A \).
- **Union**: The union of two sets \( X \) and \( Y \), denoted \( X \cup Y \), includes all elements that are in either \( X \) or \( Y \) or in both.

### Step 2: Find \( A - B \)
To find \( A - B \):
- Identify all elements in set \( A \).
- Remove any elements that are also in set \( B \).

### Step 3: Find \( B - A \)
To find \( B - A \):
- Identify all elements in set \( B \).
- Remove any elements that are also in set \( A \).

### Step 4: Combine the results using Union
Now, we need to take the union of the two results:
- \( (A - B) \cup (B - A) \) will include all elements that are either in \( A \) but not in \( B \) or in \( B \) but not in \( A \).

### Step 5: Visualize using a Venn Diagram
Using a Venn diagram can help visualize this:
- Draw two overlapping circles representing sets \( A \) and \( B \).
- The area that is only in circle \( A \) corresponds to \( A - B \).
- The area that is only in circle \( B \) corresponds to \( B - A \).
- The union of these two areas will be the total area that is either in \( A \) or in \( B \) but not in both.

### Step 6: Identify the correct option
Now, we need to check the given options to see which one represents \( (A - B) \cup (B - A) \):
1. **Option 1**: \( A \cup B \) - This includes the intersection, so it is incorrect.
2. **Option 2**: \( A \cup B \) - This also includes the intersection, so it is incorrect.
3. **Option 3**: \( (A \cup B) - (A \cap B) \) - This correctly represents the union of the two sets without the intersection.
4. **Option 4**: \( A - B \cap B - A \) - This is incorrect as it does not represent the union correctly.

### Conclusion
The correct answer is **Option 3**: \( (A \cup B) - (A \cap B) \).

---