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That is, how to represent set $A$ in terms of D, B, C given the equation above? Or is it possible?

N.B. To represent a set is to define this set, i.e. $A = \cdots$, not $A \subseteq \cdots$.

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    Here is a [diagram](http://i.stack.imgur.com/qgsVv.png) to accompany answers you've been already given. You can see that $D$ (the shaded area) does not depend in any way on elements in $A\cap C\setminus B$. (This is one of the areas in the picture.)2012-11-01

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If $D\cap C=\varnothing$ and $B\subseteq D\cup C$, the solutions are the sets $A$ such that $D\setminus B\subseteq A\subseteq D\cup C$. Otherwise, there is no solution.

To see this, picture $B$ as the upper half of a square, $C$ as its left half and $D$ as its right half. Then $A$ must contain the lower-right quarter and must not meet the top-left quarter.

Edit: There exists a unique solution $A$ if and only if $D\cap C=\varnothing$, $B\subseteq D\cup C$ (existence) and $D\setminus B=D\cup C$ (uniqueness). These conditions are equivalent to $B=C=\varnothing$, then the unique solution is $A=D$ (obviously).

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    Meadow: Somehow in the same vein, I am rather ba$f$fled that you still did not see fit to accept [this answer](http://math.stackexchange.com/a/182416).2012-11-01