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 two sets such that $A \cap X=B \cap X=f$ and $A \cup X=B \cup X$ for some set $X$.

To show: $A=B$

It can be seen that

$A=A \cap(A \cup X)=A \cap(B \cup X)[A \cup X=B \cup X]$

$=(A \cap B) \cup(A \cap X)[$ Distributive law $]$

$=(A \cap B) \cup \Phi[A \cap X=\Phi]$

$=A \cap B \ldots(1)$

Now, B $=B \cap(B \cup X)$

$=B \cap(A \cup X)[A \cup X=B \cup X]$

$=(B \cap A) \cup(B \cap X)[$ Distributive law $]$

$=(B \cap A) \cup \Phi[B \cap X=\Phi]$

$=B \cap A$

$=A \cap B \ldots(2)$

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