Question

If A∩B= d. then which of the following is correct?

• A. A = ϕ and B ≠ ϕ
• B. A ≠ ϕ and B = ϕ
• C. All of these
• D. A and B are disjoint sets A

Answer 1

The Correct Option is D

Step 1: Recall the definition of disjoint sets.

If \( A \cap B = \phi \), it means that sets \( A \) and \( B \) have no elements in common. Such sets are called disjoint sets.

Step 2: Analyze each option.

• (1) A and B are disjoint sets: This is true because \( A \cap B = \phi \) directly implies that \( A \) and \( B \) are disjoint.
• (2) \( A \neq \phi \) and \( B = \phi \): This is not necessarily true. For example, if \( A = \{1, 2\} \) and \( B = \{3, 4\} \), then \( A \cap B = \phi \), but neither \( A \) nor \( B \) is empty.
• (3) \( A = \phi \) and \( B \neq \phi \): This is also not necessarily true. For example, if \( A = \{1, 2\} \) and \( B = \{3, 4\} \), then \( A \cap B = \phi \), but neither \( A \) nor \( B \) is empty.
• (4) All of these: This is incorrect because options (2) and (3) are not always true.

Final Answer: The correct statement is \( \mathbf{\text{A and B are disjoint sets}} \), which corresponds to option \( \mathbf{(1)} \).