Question:
Let $A$ and $B$ be sets. If $A \cap X=B \cap X=\Phi$ and $A \cup X=B \cup X$ for some set $X$, show that $A=B$.
(Hints $A=A \cap(A \cup X), B=B \cap(B \cup X)$ and use distributive law)
Solution:
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.