Question

lf U= {1, 2, 3, 4, 5, 6, 7, 8, 9}, A= {2, 4, 6, 8} and B= {2, 3, 5, 7}. Verify that $(A \cap B)^{'} = A^{'} \cup B^{'}$

Answer 1

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Answer 2

First solving for left hand side,
(A ⋂ B)’ = U - (A ⋂ B)
First we have to find (A ⋂ B),
(A ⋂ B) = {2, 4, 6, 8} ⋂ {2, 3, 5, 7}
⇒ (A ⋂ B) = {2}
Now,
(A ⋂ B)’ = U - (A ⋂ B)
⇒ (A ⋂ B)’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2}
⇒ (A ⋂ B)’ = {1, 3, 4, 5, 6, 7, 8, 9}

Now, solving for right hand side,
A’ = U - A
A’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}
A’ = {1, 3, 5, 7, 9}
B’ = U - B
B’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 5, 7}
B’ = {1, 4, 6, 8, 9}
A’ ⋃ B’ = {1, 3, 5, 7, 9} ⋃ {1, 4, 6, 8, 9}
⇒ A’ ⋃ B’ = {1, 3, 4, 5, 6, 7, 8, 9}

∴ LHS = RHS, Hence verified.