How do I complete this identity? I'm given this identity:
$(A \bigtriangleup B) \cap C = (C \backslash A) \bigtriangleup ???$
Any hints on what I might use to fill the $???$ ?
The set defined by left part of the identity is the yellow area and the one from the right side (the incomplete one) is the blue area. Here is what I tried:
$(x \in A \vee x \in B) \wedge \neg(x \in A \wedge x \in B) \wedge x \in C = (x \in C \wedge x \notin A) \vee ...$
I can't figure out how to continue the right part... any ideas?

 A: I'll use an alternate definition of the symmetric difference. Perhaps this will fal out for you. Be patient! You can do this without knowing ahead of time what exactly you're after. $$\begin{align} x\in (A\triangle B) \cap C  &\iff [(x \in A \land x\notin B)\lor (x\in B\land x\notin A)]\land x\in C\\ \\ & \iff [x \in A \land x\notin B \land x\in C]\lor [x\in B\land x\notin A \land x\in C] \\ \\ & \iff [x\in A \lor (x\in B \land x\notin A\land x\in C)] \\ &\qquad \land [x\notin B \lor (x\in B \land x\notin A \land x\in C)] \\ &\qquad \land[x\in C \lor (x\in B \land x\notin A \land x\in C)]\\ \\  \quad \cdots\end{align}$$
The final conjunct, immediately above, reduces to $\land (x\in C)$.
A: Hint: $(U \Delta V) \Delta U = V$.  
A: Look at the figures. The $\bigtriangleup$ operation gives you everything that is in one but not in both operands. To get from the blue to the yellow area, you need to  (1) take something away (the lower part) and (2) add something (what is yellow in the left panel but not blue in the right one). Can you express (1) and (2) independently? Can you express their union? This is what you need to "$\bigtriangleup$" with the blue area, i.e. the right hand side.
