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

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 

Answer
Let x ∈ A
⇒ x ∈ (A or X)
⇒ x ∈ (A ∪ X)
Given that A ∪ X = B ∪ X
⇒ x ∈ (B ∪ X)
⇒ x ∈ B or x ∈ X
Given that A ∩ X = Φ and x ∈ A. So that x ∉ X.
Therefore, A ⊂ B ... (1)
Similarly take
⇒ y ∈ B
⇒ y ∈ (B or X)
⇒ y ∈ (B ∪ X)
Given that A ∪ X = B ∪ X
⇒ y ∈ (A ∪ X)
⇒ y ∈ A or y ∈ X
Given that B ∩ X = Φ and y ∈ B. So that y ∉ X.
Therefore, B ⊂ A ... (2)
From (1) and (2) we get A = B