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.

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
$$A \setminus (B \setminus C) = \{x : x \in A \text{ and } (x \not \in B \text{ or } x \in C) \} \} $$
