Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3,

1.	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’
Answer» 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.

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’

Answer»

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.