Question

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
(i) \((A ∪ B)' = A ' ∩ B'\) (ii) \((A ∩ B)' = A' ∪ B\)

Answer 1

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {2, 4, 6, 8}, B = {2, 3, 5, 7}

(i) (\(A ∪ B)' =\) {2,3,4,5,6,7,8}' = {1,9}
\(A' ∩ B' =\) {1,3,5,7,9} \(\cap\) (1,4,6,8,9) = {1,9}
\(∴ (A ∪ B)' = A' ∩ B'\)

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(ii) \((A ∩ B)' =\) {2}\('\) = {1,3,4,5,6,7,8,9}
\(A' ∪ B' =\) {1,3,5,7,9} \(\cup\) {1,4,6,8,9} = {1,3,4,5,6,7,8,9}
\(∴ (A ∩ B)' = A' ∪ B' \)