Let A, B, C be arbitrary sets. Solve this system of equations, and find necessary and sufficient conditions for existence and uniqueness of the solution:
$A \cup X = B\cap X$
$A \cap X = C \cup X$
Let A, B, C be arbitrary sets. Solve this system of equations, and find necessary and sufficient conditions for existence and uniqueness of the solution:
$A \cup X = B\cap X$
$A \cap X = C \cup X$
For any sets $U,V$, we have $U\cap V\subseteq U \subseteq U\cup V$.
So, from the first one: $B\cap X\subseteq X\subseteq A\cup X = B\cap X $ so all these must be $=$, and from the second one: $A\cap X\subseteq X\subseteq C\cup X = A\cap X$ ... can you continue?