Question

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5, 6} and B = {2, 3, 4} then prove that: 

(i) (A ∪B)’ = A’ ∩ B’ 

(ii) (A ∩ B)’ = A’ ∪ B’

Answer 1

A’ = U – A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {1,2, 3, 4, 5, 6} = {7, 8, 9, 10} 

B’ = U – B = {1,2, 3, 4, 5, 6, 7, 8, 9, 10} – {2, 3, 4} = {1, 5, 6, 7, 8, 9, 10} 

(i) (A ∪B) = {1, 2, 3, 4, 5, 6} ∪ {2, 3, 4} = {1, 2, 3, 4, 5, 6} 

(A ∪B)’ = U – (A ∪ B) 

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {1, 2, 3, 4, 5, 6} = {7, 8, 9, 10} 

and A’ ∩ B’ = {7, 8, 9, 10} ∩ {1, 5, 6, 7, 8, 9, 10} = {7, 8, 9, 10} 

(A ∪B)’ = A’ ∩ B’ 

(ii) A ∩ B = {1, 2, 3, 4, 5, 6,} ∩ {2, 3, 4} = {2, 3, 4}

(A ∩ B)’ = U – (A ∩ B) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {2, 3, 4} = {1, 5, 6, 7, 8, 9, 10} 

A’ ∪ B’ = {7, 8, 9, 10} ∪ {1, 5, 6, 7, 8, 9, 10} = {1, 5, 6, 7, 8, 9, 10} 

(A ∩ B)’ = A’ ∪ B’