How would you write $A\setminus(B\setminus C)$? I know that $(A \setminus B) = \{x: x \in A$ and $x \not \in B)\}$, but am not sure how to write $A\setminus(B\setminus C)$ in the same form.
 A: Consider that $~x\notin B~$ is $~\neg(x\in B)~$.
So you can write: $~A\setminus B= \{x: x\in A\wedge \neg (x\in B)\}~$.
Likewise you can write: $~A\setminus (B\setminus C) = \{x: x\in A \wedge \neg(x\in B\wedge \neg (x\in C)) \}~$.
So, with a little dual negations: $$A\setminus (B\setminus C) = \{x: x\in A \wedge (x\notin B\vee x\in C) \}$$
A: $$\begin{align*}
A \setminus (B \setminus C)
&= \{x\mid x \in A \land x \not\in (B \setminus C)\} \\
&= \{x\mid x \in A \land （x \in C \lor x \not\in B)\}
\end{align*}$$
A: $A\setminus(B\setminus C)=\{x:x\in A\wedge(x\not\in B\vee x\in C)\}$
A: $A - (B - C)$ is the set of all elements in $A$ which are not in $B - C$. $B - C$ is the set of all elements in $B$ that are not in $C$. Thus $A$ consists of all $x$ such that $x \in A$, and $\neg (x \in B\ \text{and}\ x \not \in C)$. You can make this nicer using Boole's identities. I leave this for you to finish (I assume this is a homework problem, so its good practice to complete it yourself). 
A: Let $D = (B \setminus C)$. Then 
$$A \setminus (B \setminus C) = A \setminus D = \{x : x \in A \text{ and } x \not \in D \}$$ 
Additionally, we have 
$$D = (B \setminus C) = \{y : y \in B \text{ and } y \not \in C \}$$
Substituting for $D$, we get 
$$A \setminus (B \setminus C) = \{x : x \in A \text{ and } x \not \in \{y : y \in B \text{ and } y \not \in C \} \} $$
Applying definitions of element, etc. we get
$$A \setminus (B \setminus C) = \{x : x \in A \text{ and } (x \not \in B \text{ or } x \in C) \} \} $$
