Find (i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′ (vi) (B – C)′

Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 }.
Find (i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′ (vi) (B – C)′

Answer
U ={1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4}
(i) A’ 
⇒U – A
⇒{1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4}
⇒{ 5, 6, 7, 8, 9}
(ii) B’
⇒ U – B
⇒{1, 2, 3, 4, 5, 6, 7, 8, 9} - { 2, 4, 6, 8 }
⇒{1, 3, 5, 7, 9 }

(iii) (A ∪ C)′
Plug the value of A and C we get
⇒ ({1, 2, 3, 4} ∪ { 3, 4, 5, 6 })′
⇒ {1, 2, 3, 4, 5, 6}’
⇒ U - {1, 2, 3, 4, 5, 6}
Plug the value of U we get
⇒{1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 5, 6}
⇒ {7, 8, 9 }

(iv) (A ∪ B)′
Plug the value of A and B we get
⇒ ({1, 2, 3, 4} ∪ { 2, 4, 6, 8 })′
⇒ {1, 2, 3, 4, 6, 8}’
⇒ U - {1 ,3, 5, 7, 9}
Plug the value of U we get
⇒ {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 6, 8}
⇒ { 5, 7, 9 )
(v) (A′)′
⇒( A’)’
⇒(U – A)’
⇒({1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4})’
⇒{ 5, 6, 7, 8, 9}’
⇒ U - { 5, 6, 7, 8, 9}
⇒ { 1, 2, 3, 4 }
(vi) (B – C)′ 
= ({ 2, 4, 6, 8 } - { 3, 4, 5, 6 })’
= {2, 8}’
= U - {2, 8}
={1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 8}
={ 1, 3, 4, 5, 6, 7, 9 }