To show:

A = B It can be seen that

A = A ∩ (A ∪ X) = A ∩ (B ∪ X)

[A ∪ X = B ∪ X] = (A ∩ B) ∪ (A ∩ X)

[Distributive law] = (A ∩ B) ∪ Φ [A ∩ X = Φ] = A ∩ B …………………………………………………………….. (1)

Now,

B = B ∩ (B ∪ X) = B ∩ (A ∪ X) [A ∪ X = B ∪ X] = (B ∩ A) ∪ (B ∩ X)

[Distributive law] = (B ∩ A) ∪ Φ [B ∩ X = Φ] 10 = B ∩ A = A ∩ B …………………………………………………………… (2) Hence,

from (1) and (2), we obtain A = B.