Question

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify the following:
B = (A ∩ B) ∪ (A'∩ B)

Answer 1

A = {1, 2, 3, 4},
B = {3, 4, 5, 6},
C = {4, 5, 6, 7, 8}
X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B = {3, 4, 5, 6}                                ...(i)
(A ∩ B) = {3, 4}
A' = {5, 6, 7, 8, 9, 10}
A' ∩ B = {5, 6}
∴ (A ∩ B) ∪ (A' ∩ B) = {3, 4, 5, 6}   ...(ii)
From (i) and (ii), we get
B = (A ∩ B) ∪ (A' ∩ B)