Let $A$ and $B$ be finite sets such that $A \subseteq B$. Then, what is the value of the expression:
$$ \sum \limits_{C:A \subseteq C \subseteq B} (-1)^{\mid C \setminus A \mid,}$$
Where $C \setminus A=\{x \in C : x \notin A \}$?
- Always $0$
- Always $1$
- $0$ if $A=B$ and $1$ otherwise
- $1$ if $A=B$ and $0$ otherwise
- Depends on the size of the universe