Given that

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

A = {1, 2, 3, 4}

B = {2, 4, 6, 8}

C = {3, 4, 5, 6}

(i) We have to find complement of A, which is given by (U - A)

⇒ A’ = U - A

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

⇒ A’ = {5, 6, 7, 8, 9}

(ii) B’ = U - B

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

⇒ B’ = {1, 3, 5, 7, 9}

(iii) (A ⋃ C)’ = U - (A ⋃ C)

First we find (A ⋃ C)

⇒ (A ⋃ C) = {1, 2, 3, 4} ⋃ {3, 4, 5, 6}

⇒ (A ⋃ C) = {1, 2, 3, 4, 5, 6}

Now,

(A ⋃ C)’ = U - (A ⋃ C)

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

⇒ (A ⋃ C)’ = {7, 8, 9}

(iv) (A ⋃ B)’ = U - (A ⋃ B)

First we find (A ⋃ B),

(A ⋃ B) = {1, 2, 3, 4} ⋃ {2, 4, 6, 8}

⇒ (A ⋃ B) = {1, 2, 3, 4, 6, 8}

Now,

(A ⋃ B)’ = U - (A ⋃ B)

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

⇒ (A ⋃ B)’ = {5, 7, 9}

(v)We want to find complement of A’, which is given by U - A’

From part (i), we have A’ = {5, 6, 7, 8, 9}

⇒ (A’)’ = U - A’

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

⇒ (A’)’ = {1, 2, 3, 4} = A

We conclude that (A’)’ = A.

(vi) (B - C)’ = U - (B - C)

First we find (B - C),

⇒ (B - C) = {2, 4, 6, 8} - {3, 4, 5, 6}

⇒ (B - C) = {2, 8}

Now,

(B - C)’ = U - (B - C)

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

⇒ (B - C)’ = {1, 3, 4, 5, 6, 7, 9}