`(A-B)cup(B-A)=?`

Options

A. `(AcupB)cup(A-B)`
B. `(AcupB)cup(AcapB)`
C. `(AcupB)-(AcapB)`
D. `(A-B)cap(A-B)`

Explanation

To solve the question \((A - B) \cup (B - A)\), we will follow these steps:

### Step 1: Understand the Definitions
- **Set A**: A collection of elements.
- **Set B**: Another collection of elements.
- **Set Difference**: \(A - B\) is the set of elements that are in A but not in B. Similarly, \(B - A\) is the set of elements that are in B but not in A.
- **Union**: The union of two sets \(X\) and \(Y\), denoted \(X \cup Y\), is the set of elements that are in either \(X\), \(Y\), or both.

### Step 2: Find \(A - B\)
- The set \(A - B\) includes all elements that are in set A but not in set B.

### Step 3: Find \(B - A\)
- The set \(B - A\) includes all elements that are in set B but not in set A.

### Step 4: Find the Union \((A - B) \cup (B - A)\)
- The union of \(A - B\) and \(B - A\) includes all elements that are either in \(A - B\) or in \(B - A\). This means we are collecting all elements that are in A but not in B, along with all elements that are in B but not in A.

### Step 5: Result Interpretation
- The union \((A - B) \cup (B - A)\) represents all elements that are in either set A or set B, but not in both. This is also known as the symmetric difference of sets A and B.

### Step 6: Final Expression
- We can express the result as:
\[ (A - B) \cup (B - A) = A \cup B - A \cap B \]
This means the union of the two sets minus the intersection (common elements) of A and B.

### Final Answer
\[ (A - B) \cup (B - A) = A \cup B - A \cap B \]

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