Let $X$ be a vector space and let $A,B\subset X$. I want to know if $X\setminus (A-B) = (X\setminus A)-B$.
I'm using the following definition: $A-B = \{a-b:\ a\in A, b\in B\}$.
My thoughts: If $x\in (X\setminus A)-B$, then exists $w\notin A$ and $b\in B$ such that $x=w-b$. I want to say it's clear that $x\notin A-B$, but I'm not sure. Maybe it's possible to exists some $w\notin A$ and $b\in B$ such that $w-b\in A-B$. For instance, we may have $w=a+b'$, for $a\in A, b'\in B$, such that $a+b'\notin A$ and $b'-b\in B$, then $w-b = a+(b'-b)\in A-B$.
The other inclusion looks even more difficult to prove. Could someone please show me how this is true or show me how this is false? In the case the equality is not true, can we guarantee some inclusion?
Thanks.
We have that $$ \begin{align} x \in X \setminus (A-B) &\iff \forall a\in A, b \in B: a-b \ne x \\ &\iff \tag 1(x+ B) \cap A = \emptyset\\ &\iff x+ B \subseteq X \setminus A \end{align}$$ and $$ \tag 2 x \in (X \setminus A) - B \iff \exists b\in B: x+b \in X \setminus A \iff (x + B) \cap (X\setminus A) \ne \emptyset $$ If the sets in question are non-empty (1) implies (2), but the contrary is wrong.
An example is given by $A = B = (-1,1)$ in $X = \mathbf R$, then $A- B = (-2,2)$, and $$(X\setminus A) - B = \mathbf R\setminus \{0\} \ne \mathbf R\setminus (-2,2) $$