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’

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’

Philoid · Accepted Answer

(i) First solving for left hand side,

(A ⋃ B)’ = U - (A ⋃ B)

First we find (A ⋃ B),

(A ⋃ B) = {2, 4, 6, 8} ⋃ {2, 3, 5, 7}

⇒ (A ⋃ B) = {2, 3, 4, 5, 6, 7, 8}

Now,

(A ⋃ B)’ = U - (A ⋃ B)

⇒ (A ⋃ B)’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 4, 5, 6, 7, 8}

⇒ (A ⋃ B)’ = {1, 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, 9}

∴ LHS = RHS, Hence verified.

(ii) First solving for left hand side,

(A ⋂ B)’ = U - (A ⋂ B)

First we 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}