Hello I already asked this question here Left quotient of a regular languagebut I asked from anonymous profile and now I couldn't comment on this. So I ask here for more details...Question:Show that for any language $L  ^*$ and any DFA $A = \langle \Sigma, Q, q_0, \delta, F \rangle$,  the left quotient $L \diagdown L (A)$ is a union of languages $L_q = \{v | \delta(q,v) \in F\}$ for selected states $q \in Q$, and explain what are those selected states.As the answer I got this:By definition,$$L \diagdown L(A) = \bigcup_{x \in L} \{ w : \delta(q_0,xw) \in F \}.$$The idea now is to use the identity$$\delta(q_0,xw) = \delta(\delta(q_0,x),w).$$You take it from here.But I still can't see why from the answer? Could you please show me some simple example? I cannot figure this out from this answer. Thanks.
Left quotient of a regular language  follow-up
formal languages;regular languages;finite automata