I'm given an open set O, a closed set F, and an arbitrary set E;
where E is a subset of O and F is a subset of E.
Is it true that O~F = (E~F) union (O~E)?
I assume that by $\displaystyle O\sim F$ that you mean set difference.
That is $\displaystyle O\setminus F=O\cap F^c$, if so then yes $\displaystyle O\setminus F=(E\setminus F)\cup(O\setminus E)$.