Question

The set \( (A \setminus B) \cup (B \setminus A) \) is equal to

• A. \( [A \setminus (A \cap B)] \cap [B \setminus (A \cap B)] \)
• B. \( (A \cup B) \setminus (A \cap B) \)
• C. \( A \setminus (A \cap B) \)
• D. \( \overline{A \cap B} \setminus (A \cup B) \)
• E. \( (\overline{A} \cup B) \setminus (\overline{A \cup B}) \)

Hint:

The expression \( (A \setminus B) \cup (B \setminus A) \) is always the symmetric difference: elements belonging to exactly one set.

Answer 1

The Correct Option is B

Concept: The expression \( (A \setminus B) \cup (B \setminus A) \) represents elements that belong to exactly one of the sets. This is called the symmetric difference.

Step 1: Understand each term
\[ A \setminus B = \text{elements in } A \text{ but not in } B \] \[ B \setminus A = \text{elements in } B \text{ but not in } A \]

Step 2: Combine the sets
\[ (A \setminus B) \cup (B \setminus A) \] This includes elements present in either set but not common.

Step 3: Express using standard identities
We know: \[ A \setminus B = A \cap B^c \] \[ B \setminus A = B \cap A^c \] So, \[ (A \cap B^c) \cup (B \cap A^c) \]

Step 4: Use identity
This is equal to: \[ (A \cup B) \setminus (A \cap B) \]

Step 5: Final answer
\[ \boxed{(A \cup B) \setminus (A \cap B)} \]