Question

If \( A = \{1, 2, 4\} \), \( B = \{2, 4, 5\} \), \( C = \{2, 5\} \), then \[ (A - B) \cap (B - C) = \]

• A. \( \{2, 4, 5\} \)
• B. \( \{1, 2, 4, 5\} \)
• C. \( \emptyset \)
• D. \( \{4, 5\} \)

Hint:

When solving set operations such as differences and intersections, make sure to carefully follow the definitions of these operations and apply them step by step.

Answer 1

The Correct Option is C

Step 1: Find \( A - B \).
The set \( A - B \) represents the elements that are in \( A \) but not in \( B \). So:
\[ A = \{1, 2, 4\}, \quad B = \{2, 4, 5\} \]
Thus, \[ A - B = \{1\} \]

Step 2: Find \( B - C \).
The set \( B - C \) represents the elements that are in \( B \) but not in \( C \). So:
\[ B = \{2, 4, 5\}, \quad C = \{2, 5\} \] Thus, \[ B - C = \{4\} \]

Step 3: Find the intersection \( (A - B) \cap (B - C) \).
Now, we find the intersection of \( A - B \) and \( B - C \):
\[ A - B = \{1\}, \quad B - C = \{4\} \]
Thus, the intersection of these two sets is: \[ (A - B) \cap (B - C) = \{1\} \cap \{4\} = \emptyset \]

Step 4: Conclusion.
Therefore, the value of \( (A - B) \cap (B - C) \) is \( \emptyset \), which corresponds to option (C).