Question

Let A and B be sets. If \(A ∩ X = B ∩ X = \phi\) 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] = \((A ∩ B) ∪ \phi [A ∩ X = \phi] \)
\(= A ∩ B\) …………………………………………………………….. (1) 

Now, \(B = B ∩ (B ∪ X) \)
\(= B ∩ (A ∪ X) [A ∪ X = B ∪ X] \)
\(= (B ∩ A) ∪ (B ∩ X)\) [Distributive law] 
\(= (B ∩ A) ∪ \phi [B ∩ X = \phi]\)
\(= B ∩ A \)
\(= A ∩ B\) …………………………………………………………… (2) 

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