Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that:
(A ∪ B)’ = A’ ∩ B’
A ∪ B = {x: x ϵ A or x ϵ B}
= {2, 3, 4, 5, 6, 7, 8}
(A∪B)’ means Complement of (A∪B) with respect to universal set U.
So, (A∪B)’ = U – (A∪B)’
U – ( A∪B)’ is defined as {x ϵ U : x ∉ (A∪B)’}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
(A∪B)’ = {2, 3, 4, 5, 6, 7, 8}
U – ( A∪B)’ = {1, 9}
Now
A’ means Complement of A with respect to universal set U.
So, A’ = U – A
U – A is defined as {x ϵ U : x ∉ A}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {2, 4, 6, 8}
A’ = {1, 3, 5, 7, 9}
B’ means Complement of B with respect to universal set U.
So, B’ = U – B
U – B is defined as {x ϵ U : x ∉ B}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
B = {2, 3, 5, 7}.
B’ = {1, 4, 6, 8, 9}
A’ ∩ B’ = = {x:x ϵ A’ and x ϵ C’}.
= {1, 9}
Hence verified.