Show that :
`(AuuB)-(AcapB)=(A-B)uu(B-A)`.

Answer

To show that \((A \cup B) - (A \cap B) = (A - B) \cup (B - A)\), we can use set operations and properties. Let's break it down step by step.

### Step 1: Understand the Left-Hand Side (LHS)
The left-hand side of the equation is \((A \cup B) - (A \cap B)\). This represents the elements that are in either set \(A\) or set \(B\) but not in both.

### Step 2: Rewrite the LHS
Using the definition of set difference, we can express the LHS as:
\[ (A \cup B) - (A \cap B) = \{ x \mid x \in (A \cup B) \text{ and } x \notin (A \cap B) \} \]

### Step 3: Analyze the Elements
The elements in \(A \cup B\) are those in \(A\), in \(B\), or in both. The elements in \(A \cap B\) are those that are in both \(A\) and \(B\). Therefore, the elements that are in \(A \cup B\) but not in \(A \cap B\) are those that are exclusively in \(A\) or exclusively in \(B\).

### Step 4: Rewrite the LHS Using Set Differences
Thus, we can express the LHS as:
\[ (A \cup B) - (A \cap B) = (A - B) \cup (B - A) \]
This is because:
- \(A - B\) includes elements in \(A\) that are not in \(B\).
- \(B - A\) includes elements in \(B\) that are not in \(A\).

### Step 5: Understand the Right-Hand Side (RHS)
The right-hand side of the equation is \((A - B) \cup (B - A)\). This represents the elements that are either in \(A\) but not in \(B\) or in \(B\) but not in \(A\).

### Step 6: Conclusion
Since both the LHS and RHS represent the same set of elements, we conclude that:
\[ (A \cup B) - (A \cap B) = (A - B) \cup (B - A) \]
Thus, we have shown that the two sides of the equation are equal.