Straight Lines

Question
CBSEENMA11014641

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 whether the following is true or false (give reason for your answer):
A and B' are mutually exclusive.

Solution

Two dice are rolled:
rightwards double arrow                S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4),
                           (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2),
                           (4,3), (4,4), (4, 5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6),
                           (6,1), (6,2),(6,3), (6,4), (6,5), (6,6)}                                                  ...(I)
A: Getting an even number on the first die.
rightwards double arrow                 A = {(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)}                              ...(II)
    
B: Getting an odd number on the first die.  
rightwards double arrow                  B = {(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)}                             ...(III)
 
C: Getting a sum less or equal than space 5 on the two dice.
rightwards double arrow                       C = {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)          ...(IV)
From (I) and (III)
                      B' = S - B
                          = {(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)}
                          = A                                                                 [By using part (iii)]
              straight A intersection straight B apostrophe space equals space straight A intersection straight A space equals space straight A not equal to straight ϕ
∴ A and B' are not mutually exclusive.
Hence, the statement is false.