Question

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’

Answer 1

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.