Left quotient finite automata

Question

This is very unclear to me. I understand what the left quotient is, however, can someone explain this:

A={0,101} B=01,10,001}

then how is A\B={1,01}

I mean where are these strings extracted from? I dont see them in A. How can I appeand 01 to anything is B to get something in B?

Answer 1

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 $0 \in A$.