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 $A = B$. (Hint: Use $A = A \cap (A \cup X)$,$B = B \cap (B \cup X)$ and the Distributive law.)

Given that $A \cap X = \phi$,$B \cap X = \phi$,and $A \cup X = B \cup X$ for some set $X$.
To show: $A = B$.
We know that $A = A \cap (A \cup X)$.
Substituting $A \cup X = B \cup X$,we get:
$A = A \cap (B \cup X)$.
Using the Distributive law,$A = (A \cap B) \cup (A \cap X)$.
Since $A \cap X = \phi$,we have $A = (A \cap B) \cup \phi = A \cap B$ ... $(1)$.
Similarly,$B = B \cap (B \cup X)$.
Substituting $B \cup X = A \cup X$,we get:
$B = B \cap (A \cup X)$.
Using the Distributive law,$B = (B \cap A) \cup (B \cap X)$.
Since $B \cap X = \phi$,we have $B = (B \cap A) \cup \phi = B \cap A = A \cap B$ ... $(2)$.
From $(1)$ and $(2)$,we conclude that $A = B$.