<p>Which of the following statement is false?</p><p>(a) A and B are mutually exclusive</p><p>(b) A and B are mutually exclusive and exhaustive</p><p>(c) \(A = B'\)</p><p>(d) A and C are mutually exclusive</p>
Step-by-Step Solution
Key Concept: Two events are mutually exclusive if their intersection is empty. Check which pair has non-empty intersection.
<p><strong>Step 1:</strong> If two dice are thrown, then total number of possible outcomes, $S = 6 \times 6 = 36$</p><p><strong>Step 2:</strong> $A$ = getting an even number on the first die = \{(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (4, 1), (4, 2), (4,3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6,6)\}</p><p><strong>Step 3:</strong> $B$ = getting an odd number on the first die = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3,3), (3, 4), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5,6)\}</p><p><strong>Step 4:</strong> $C = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)\}$</p><p><strong>Step 5:</strong> Since $A \cap C \neq \emptyset$ (they share common elements like (2,1), (2,2), (2,3), (4,1)), statement (d) is false.</p><p>∴ Answer is D.</p>
Correct Answer: D