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Question
From the given diagram, find:
(i) A’
(ii) B’
(iii) A' ∪ B'
(iv) (A ∩ B)'
Is A' ∪ B' = (A ∩ B)' ?
Also, verify if A' ∪ B' = (A ∩ B)'.
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Solution
(i) A = {1, 3, 4, 6}
A' = {2, 5, 7, 8, 9, 10}
(ii) B = {1, 2, 5}
∴ B' = {3, 4, 6, 7, 8, 9, 10}
(iii) A' ∪ B' = {2, 5, 7, 8, 9, 10} ∪ {3, 4, 6, 7, 8, 9, 10}
= {2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv) A ∩ B = {1, 3, 4, 6} ∩ {1, 2, 5}
= {1}
∴ (A ∩ B)' = {2, 3, 4, 5, 6, 7, 8, 9, 10}
From Part (iii) and Part (iv) we conclude
A' ∪ B' = (A ∩ B)'
Now A ∩ B = {2, 5, 7, 8, 9, 10} ∩ {3, 4, 6, 7, 8, 9, 10}
⇒ A' ∪ B' = {7, 8, 9, 10} ...(I)
Now A ∪ B = {1, 3, 4, 6} ∪ {1, 2, 5}
= {1, 2, 3, 4, 5, 6}
∴ (A ∩ B)' = {7, 8, 9, 10} ...(II)
From I and II we conclude
A' ∪ B' = (A ∩ B)'
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