which one of the following is `(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 problem of finding which one of the following is equal to \( (A - B) \cup (B - A) \), we can follow these steps:

### Step 1: Understand the Sets
- **Set A**: Represents one group of elements.
- **Set B**: Represents another group of elements.
- **\( A - B \)**: This represents the elements that are in set A but not in set B.
- **\( B - A \)**: This represents the elements that are in set B but not in set A.

### Step 2: Write the Expression
The expression we need to evaluate is:
\[ (A - B) \cup (B - A) \]
This means we are looking for the union of the elements that are only in A and the elements that are only in B.

### Step 3: Visualize with a Venn Diagram
To better understand this expression, we can use a Venn diagram:
- Draw two overlapping circles, one for set A and one for set B.
- Shade the part of circle A that does not overlap with circle B (this is \( A - B \)).
- Shade the part of circle B that does not overlap with circle A (this is \( B - A \)).
- The total shaded area represents \( (A - B) \cup (B - A) \).

### Step 4: Identify What is Left Out
The parts that are not shaded in the Venn diagram are the elements that are in both A and B (the intersection \( A \cap B \)). Therefore, the union of \( A - B \) and \( B - A \) excludes the intersection.

### Step 5: Express in Terms of Union and Intersection
From the Venn diagram, we can conclude that:
\[ (A - B) \cup (B - A) = (A \cup B) - (A \cap B) \]
This means that the union of the two sets minus their intersection gives us the desired result.

### Step 6: Conclusion
Thus, the expression \( (A - B) \cup (B - A) \) can be rewritten as:
\[ A \cup B - A \cap B \]
Now, we can check the options provided in the question to find the equivalent expression.