How do you write, $(A∩B)\setminus (A∩C)$. How do you write, $(A\cap B)\setminus (A\cap C)$. I know $(A \cap B) = \{x: x \in A$ and $x \in B\}$, and $(A \setminus B) = \{x: x \in A$ and $x \not \in B)\}$, but I don't know how to combine these facts into one.
You are taking $A\cap B$ and excluding something from it. Anything you exclude from $A\cap B$ is automatically in $A$. So at that point, writing that you are excluding $A\cap C$ is a little redundant. You can just say you are excluding $C$. So you have the same thing as $$(A\cap B)\setminus C$$ It's hard to imagine anything simpler exists, since this references the three sets involved each precisely once. This is the same as $$A\cap(B\setminus C)$$ so you could just write $$A\cap B\setminus C$$

If you want set-builder notation, this is $$\{x\mid x\in A \text{ and } x\in B \text{ and } x\not\in C\}$$
You can learn a few simple laws of set theory in one week and use them always.  

$(A\cap B)\setminus (A\cap C)=[\color{green}{\text{by Boby's comment}}]$  

$(A\cap B)\cap (A\cap C)^c=$ [by De Morgan's laws]
$\color{red}{(A\cap B)}\cap (A^c\cup C^c)=$ [by this question] 

$(\color{red}{A\cap B}\cap A^c)\cup (\color{red}{A\cap B}\cap C^c)=$  

$(A\cap B\cap C^c)=[\color{green}{\text{by Boby's comment}}]$  

$(A\cap B)\setminus C$