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Question

Let set U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, set A = {2, 4, 6, 8}, and set B = {1, 3, 5, 7, 9}. Prove that (A ∩ B)’ = A’ ∪ B.’

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Solution

(A ∩ B)’ = A’ ∪ B.’ We will evaluate both the left-hand side and the right-hand side separately. Left-Hand Side: (A ∩ B)’ Let’s first the intersection of A and B given as: (A ∩ B) So, (A ∩ B) = {2, 4, 6, 8} ∩ {1, 3, 5, 7, 9} (A ∩ B) = { } Now, let’s find the complement of the intersection, which is given as: (A ∩ B)’ = U − (A ∩ B) (A ∩ B)’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} − { } (A ∩ B)’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Now, let’s evaluate the right-hand side. Right-Hand Side: A’ ∩ B’ Let’s first find the complements: The complement of A is: A’ = U − A A’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} − {2, 4, 6, 8} A’ = {0, 1, 3, 5, 7, 9, 10} The complement of B is: B’ = U − B B’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} − {1, 3, 5, 7, 9} B’ = {0, 2, 4, 6, 8, 10} Now, the union of the complements is given as: A’ ∪ B’ = {0, 1, 3, 5, 7, 9, 10} ∪ {0, 2, 4, 6, 8, 10} A’ ∪ B’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} This proves that left-hand side = right-hand side and hence: (A ∩ B)’ = A’ ∩ B.’

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