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Let $L$ be a regular language with alphabet $\Sigma = \{a,b,c\}$.

Prove that the following language is regular:

$\{w | w \in L \text{ and } w \text{ starts with } abc \}$.

I wonder what proof strategy I can use to prove this.

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  • $\begingroup$ Welcome to Computer Science! What have you tried? Where did you get stuck? We do not want to just hand you the solution; we want you to gain understanding. However, as it is we do not know what your underlying problem is, so we can not begin to help. See here for tips on asking questions about exercise problems. If you are uncertain how to improve your question, why not ask around in Computer Science Chat? $\endgroup$ – Raphael Jun 5 '17 at 21:25
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Use closure properties of regular languages.

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  • $\begingroup$ $\{w | w \in L \text{ and } w \text{ starts with } abc \} = L \cap abc(a|b|c)^*$. And regular languages are closed under intersection. Am I right? $\endgroup$ – fornit Jun 5 '17 at 21:21
  • $\begingroup$ Right, that's the simplest proof. $\endgroup$ – Yuval Filmus Jun 5 '17 at 21:40

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