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

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  • $\begingroup$ Yes it is regular. $\endgroup$
    – Nathaniel
    Dec 6, 2021 at 18:45
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    $\begingroup$ (Now if you want a better answer, you should try to improve your post: formatting, what you tried, your thoughts about the problem, …) $\endgroup$
    – Nathaniel
    Dec 6, 2021 at 18:47
  • $\begingroup$ but if i dont know anything. please help to prove it that languagee is regular $\endgroup$
    – user144938
    Dec 6, 2021 at 19:38
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    $\begingroup$ Here's an introduction to formatting math on this site: math.meta.stackexchange.com/questions/5020/… $\endgroup$
    – rici
    Dec 6, 2021 at 20:00
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    $\begingroup$ Does this answer your question? Closure against right quotient with a fixed language $\endgroup$
    – xskxzr
    Dec 7, 2021 at 8:31

1 Answer 1

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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.

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