Question

If A = {1, 3, 5, 7, 9}, B = {x : x is a composite number and x < 12} and C = {x : x ∈ N and 6 < x < 10} then verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

Answer 1

Given, A = {1, 3, 5, 7, 9}, B = {4, 6, 8, 9, 10} and C = {6, 7, 8, 9} 

B ∩ C = {4, 6, 8, 9, 10} ∩ {6, 7, 8, 9} = {6, 8, 9} 

A ∪ (B ∩ C) = {1, 3, 5, 6, 7, 8, 9} … (1) 

Then (A ∪ B) = {1, 3, 5, 7, 9} ∪ {4, 6, 8, 9, 10} = {1, 3, 4, 5, 6, 7, 8, 9, 10} 

(A ∪ C) = {1, 3, 5, 7, 9} ∪ {6, 7, 8, 9} = {1,3, 5, 6, 7, 8, 9} 

(A ∪ B) ∩ (A ∪ C) = {1, 3, 4, 5, 6, 7, 8, 9, 10} ∩ {1, 3, 5, 6, 7, 8, 9} 

= {1, 3, 5, 6, 7, 8, 9} … (2) 

From (1) and (2), it is verified that 

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)