Left quotient finite automata
By definition, $A \backslash B$ is the set of all strings $x$ such that $ax \in B$ for some $a \in A$. In our case, $1 \in A \backslash B$ since $01 \in B$ and $0 \in A$, and $01 \in A \backslash B$ since $001 \in B$ and $0 \in A$.