I have this problem, and I don't really understand how am I supposed to do this. Could someone please help me with this? I know what left quotient is. I also know about regular, irregular languages. But still I don't know how to show this... Any help would be appreciated, thanks in advance! Also sorry if something is not correctly in English, not my first language.Problem: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. Could someone please also demonstrate on a small (but non-trivial) example, in which the $L$ would be irregular?
Left quotient of a regular language
formal languages;context free;regular languages;finite automata