# Is the right quotient of a regular language respect to another regular language a regular language?

Will the language $$\{w\in L_1\mid \exists v, wv\in L_2\}$$ be regular if $$L_1$$ and $$L_2$$ regular languages?

• Yes it is regular. Dec 6, 2021 at 18:45
• (Now if you want a better answer, you should try to improve your post: formatting, what you tried, your thoughts about the problem, …) Dec 6, 2021 at 18:47
• but if i dont know anything. please help to prove it that languagee is regular Dec 6, 2021 at 19:38
• Here's an introduction to formatting math on this site: math.meta.stackexchange.com/questions/5020/…
– rici
Dec 6, 2021 at 20:00
• Does this answer your question? Closure against right quotient with a fixed language Dec 7, 2021 at 8:31

The prefix language of $$L$$ is the set of all prefixes of strings in $$L$$, $$\{w\mid \exists v \in \Sigma^*, wv\in L\}$$. In other words, it is the right quotient of $$L$$ with $$\Sigma^*$$, so it is certainly regular if $$L$$ is regular, by closure with right quotient.
A possibly simpler way to see this is to observe that you can construct a recogniser for the prefix language of $$L$$ by constructing the minimal DFA for $$L$$ and then making all states accepting.
Your language is the intersection of $$L_1$$ with the prefix language of $$L_2$$. Since regular languages are also closed with intersection, it must be regular if $$L_1$$ and $$L_2$$ are both regular.