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

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

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.