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}
$∴ (A ∩ B)' = A' ∪ B'$