Is $B \setminus (A+x) = B \cap (A+x)^c = B \cap A^c + x$? Let $A \subset \mathbb{R}^n$ and $x \in \mathbb{R}^n$( arbitrary). Is $B \setminus(A+x) = B \cap (A+x)^c = B \cap A^c + x$?($A+x=\{a+x: a \in A \}$ and $A\setminus B = A \cap B^c$)

I do not think this is true. Let $A=B=[0,1]$, $x=0.5$. Then $A+x=[0.5,1.5]$, and $B\setminus(A+x)=[0,0.5)$. But we know $B\cap A^{c}=\emptyset$, therefore $B\cap A^{c}+x=\emptyset$. So the two sets are not equal. 
