Verify that (i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′

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′

Answer

(i) (A ∪ B)′ = A′ ∩ B′

LHS

(A ∪ B)′

= ({2, 4, 6, 8} ∪ { 2, 3, 5, 7})’
= {2, 3, 4, 5, 6, 7, 8}’
= {1, 2, 3, 4, 5, 6, 7, 8, 9 } - {2, 3, 4, 5, 6, 7, 8}
= {1, 9}
RHS
A′ ∩ B′
={2, 4, 6, 8}’ ∩{ 2, 3, 5, 7}’
= [{1, 2, 3, 4, 5, 6, 7, 8, 9 }-{2, 4, 6, 8}]∩[{1, 2, 3, 4, 5, 6, 7, 8, 9 } - { 2, 3, 5, 7}]
={1, 3, 5, 7, 9}∩{1, 4, 6, 8, 9}
= {1,9}
LHS = RHS
(ii) (A ∩ B)′ = A′ ∪ B′
(A ∩ B)′
= ({2, 4, 6, 8} ∩ { 2, 3, 5, 7})’
= {2}’
= {1, 2, 3, 4, 5, 6, 7, 8, 9 } - {2}
= {1, 3, 4, 5, 6, 7, 8, 9 }
RHS
A′ ∪ B′
={2, 4, 6, 8}’ ∪ { 2, 3, 5, 7}’
= [{1, 2, 3, 4, 5, 6, 7, 8, 9 }-{2, 4, 6, 8}]∪ [{1, 2, 3, 4, 5, 6, 7, 8, 9 } - { 2, 3, 5, 7}]
={1, 3, 5, 7, 9}∪ {1, 4, 6, 8, 9}
= {1, 3, 4, 5, 6, 7, 8, 9}
LHS = RHS