If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:

(A ∪ B)’ = A’ ∩ B’

A ∪ B = {x: x ϵ A or x ϵ B }

= {2, 3, 5, 7, 9 }

(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 = {2, 3, 5, 7, 9}

(A∪B)’ = {2, 3, 5, 7, 9 }

U–( A∪B)’ = ϕ

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 = {2, 3, 5, 7, 9}

A = {3, 7}

A’ = {2, 5, 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 = {2, 3, 5, 7, 9}

B = {2, 5, 7, 9}.

B’ = {3}

A’ ∩ B’ = = {x:x ϵ A’ and x ϵ C’ }.

= ϕ.

Hence verified.