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प्रश्न
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify the following:
(A ∩ B)' = A' ∪ B'
उत्तर
A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8},
X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
The intersection of A and B includes elements common to both A and B:
A ∩ B = {3, 4}
The complement of A∩B includes all elements in the universal set X that are not in A∩B:
∴ (A ∩ B)' = X − A ∩ B
∴ (A ∩ B)' = {1, 2, 5, 6, 7, 8, 9, 10} ...(i)
The complement of A includes all elements in X that are not in A:
A' = X − A
A' = {5, 6, 7, 8, 9, 10}
The complement of B includes all elements in X that are not in B:
B' = X − B
B' = {1, 2, 7, 8, 9, 10}
The union of A′ and B′ includes all elements that are in either A' or B':
∴ A' ∪ B' = {1, 2, 5, 6, 7, 8, 9, 10} ...(ii)
From (i) and (ii), we get
(A ∩ B)' = A' ∪ B'
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