Question

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 1

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.