Question

If U = {x : x ∈ Z, -3 < x ≤ 9}, A = {x : x = 2P + 1, P ∈ Z, -2 ≤ P ≤ 3}, B = {x : x = q + l, q ∈ Z, 0 ≤ q ≤ 3}, verify De Morgan’s laws for complementation.

Answer 1

Given, U = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 

A = {-3, -1, 1, 3, 5, 7} and B = {1, 2, 3, 4} 

Law (i) (A ∪ B)’ = A’ ∩ B’ 

Now, A ∪ B = {-3, -1, 1, 2, 3, 4, 5, 7} 

(A ∪ B)’ = {-2, 0, 6, 8, 9} … (1) 

Then, A’ = {-2, 0, 2, 4, 6, 8, 9) and 

B’ = {-3, -2, -1, 0, 5, 6, 7, 8, 9} 

A’ ∩ B’ = {-2, 0, 6, 8, 9} … (2)

From (1) and (2) it is verified that 

(A ∪ B)’ = A’ ∩ B’ 

Law (ii) (A ∩ B)’ = A’ ∪ B’ 

Now, A ∩ B = {1, 3} 

(A ∩ B)’ = {-3, -2, -1, 0, 2, 4, 5, 6, 7, 8, 9} … (3) 

Then, A’ ∪ B’ = {-3, -2, -1, 0, 2, 4, 5, 6, 7, 8, 9} … (4) 

From (3) and (4) it is verified that 

(A ∩ B)’ = A’ ∪ B’