Refer to question 6 above, state true of 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
(v) A and B' are mutually exclusive
(vi) A', B', C are mutually exclusive and exhaustive.

Solution:

When two dice are thrown, the sample space is given by

S = {(x, y) : x, y ∈ {1, 2, 3, 4, 5, 6} }

Hence,

      {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

      (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

S = (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)}

Accordingly,

      {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

A = (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),

      (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
 

      {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

B = (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

      (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}
 

C = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1)

       (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}

(i) It is observed that A ∩ B = Φ

Therefore, A and B are mutually exclusive.

Thus, the given statement is true.

(ii) It is observed that A ∩ B = Φ and A υ B = S

Therefore, A and B are mutually exclusive and exhaustive.

Thus, the given statement is true.

(iii) It is observed that

      {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

A = (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),

      (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

and
                    {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

B' = S - B = (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),

                    (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

So A = B'.

Thus, the given statement is true.

(iv) It is observed that

A ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)} ≠ Φ

Therefore, A and C are not mutually exclusive.

Thus, the given statement is false.

(v) It is observed that A ∩ B' = A ∩ A = A

Therefore, A ∩ B' ≠ Φ; A and B' are not mutually exclusive.

Thus, the given statement is false.

(vi) It is observed that A' υ B' υ C = S

However,

B' ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)} ≠ Φ

Therefore, events A', B', and C are not mutually exclusive and exhaustive.

Thus, the given statement is false

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NCERT Solutions Class 11 Maths Chapter 16 Exercise 16.2 Question 7

Refer to question 6 above, state true of 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 (v) A and B' are (vi) A', B', C are mutually exclusive and exhaustive.

Summary:

(i) The given statement is true (ii) The given statement is true. (iii) The given statement is true. (iv) The given statement is false (v) The given statement is false (vi) The given statement is false