23. Refer to question 6 above, 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

(v) A and B′ are mutually exclusive.

(vi) A′, B′, C are mutually exclusive and exhaustive.

<p>&nbsp;<strong>23. (i) </strong>A ∩B = { (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </p><p> (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </p><p> (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}∩ { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </p><p> (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), </p><p> (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}</p><p>A ∩B =∅</p><p>Hence, A and B are mutually exclusive.</p><p>The given statement is true.</p><p><strong> (ii)</strong> A ∪B = { (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </p><p> (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </p><p> (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}∩ { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </p><p> (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), </p><p> (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}</p><p>= { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </p><p> (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </p><p> (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), </p><p> (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </p><p> (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), </p><p> (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}</p><p>= S</p><p>Hence, A B = S and A∩ B =∅</p><p>A and B are mutually exclusive and mutually exhaustive.</p><p>So, the give statement is true.</p><p><strong> (iii)</strong> A = { (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </p><p> (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </p><p> (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}</p><p>B = S – B = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </p><p> (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </p><p> (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), </p><p> (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </p><p> (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), </p><p> (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} – { (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>= { (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </p><p> (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </p><p> (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}</p><p>B' = A</p><p>Hence, the given relation is true.</p><p><strong> (iv)</strong> A ∩C = { (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </p><p> (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </p><p> (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}∩ { (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, (1)}</p><p>= { (2, 1), (2, 2), (2, 3), (4, 1)}≠∅</p><p>So, A∩ C≠∅</p><p>i.e., A and C are not mutually exclusive.</p><p>Hence, the given statement is false.</p><p><strong> (v)</strong> A∩ B' = A∩ A = A≠∅ { B' = A}</p><p>Hence A∩B'≠∅ i.e., A and B are not mutually exclusive.</p><p>So, the given statement is false.</p><p><strong> (vi)</strong> As A' = B</p><p>B = A</p><p>So, A'∩B = B∩A =∅</p><p>A'∩ C = B∩C = { (1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)}=∅</p><p>B'∩C = A∩C = { (2, 1), (2, 2), (2, 3), (4, 1)}=∅</p><p>Hence, A', B' and C are not mutually exclusive.</p><p>The given statement is false.</p>

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