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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:

$A \cup X = B\cap X$

$A \cap X = C \cup X$

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up vote 3 down vote accepted

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?

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Well, from these we have $X \subseteq A \subseteq X$, and that gives us $X = A$. Putting $X = A$ in previous equations, we have $A = B \cap A = C \cup A$, so $C \subseteq A \subseteq B$. Is this OK? I still don't really get that part about ''necessary and sufficient conditions'', but I'll try to wrap my head around that. –  ante.ceperic Oct 2 '12 at 12:53
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