Two dice are thrown. The events A, B and C are as follows getting an even number on the first die

Question

Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5
State true or false: (give reason for your answer)
(i) A and B are mutually exclusive
(ii) A and B are mutually exclusive and exhaustive
(iii) A = B’
(iv) A and C are mutually exclusive are mutually exclusive
(v) A and B’ are mutually exclusive.
(vi) A’, B’ and Care mutually exclusive and exhaustive.

Answer 1

(i) It is observed that A ∩ B = Φ
∴ A and B are mutually exclusive.
Thus, the given statement is true.
(ii) It is observed that A ∩ B = Φ and A ∪ B = S ∴ A and B are mutually exclusive and exhaustive.
Thus, the given statement is true.

(iii) It is observed that

Thus, the given statement is true.
(iv) It is observed that A ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)} ≠Φ
∴ A and C are not mutually exclusive.
Thus, the given statement is false.
(v) A ∩ B’ = A ∩ A = A
∵ A ∩ B’ ≠Φ
∴ A and B’ are not mutually exclusive.
Thus, the given statement is false.
(vi) It can be observed that: A’ U B’ U C = S
However, B’ ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)} ≠Φ
Therefore, events A’, B’ and Care not mutually exclusive and exhaustive.
Thus, the given statement is false.