Question

Let A and B be sets. If A ∩ X = B ∩ X = Φ and A ∪ X = B ∪ X for some set X,
show that A = B.
(Hints A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use distributive law)

Answer 1

Let A and B be two sets such that A ∩ X = B ∩ X = f and A ∪ X = B ∪ X for
some set X.
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]

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