# Prove the following language is regular using closures

How can I prove this language is regular using just closures and homomorphism? $$L = \{a_1b_1a_2b_2\dotsm a_nb_n \mid a_i \in L_1 , b_i \in L_2\}$$ and we know $L_1$ and $L_2$ are regular.

• How is $n$ defined? – adrianN Jan 5 '17 at 14:24
• n is a finite number – user64211 Jan 5 '17 at 14:40
• What did you try? Where did you get stuck? We're happy to help you understand the concepts but just solving exercises for you is unlikely to achieve that. You might find this page helpful in improving your question. Also, I suspect what adrianN is asking is: Is $n$ fixed in advance (and the same for every word in $L$), or is it free to vary? – D.W. Jan 5 '17 at 19:07
• I'm trying to understand what are the steps i am supposed to do in order to solve this kind of questions. I don't know how to begin and would love to get some help. – user64211 Jan 6 '17 at 6:28
• Try to express $L$ in terms of $L_1$ and $L_2$ using some known algebraic operator. E.g. If we knew that (random example) $L = (L_1(L_1 + L_2)) \setminus L_2L_1$ we could conclude $L$ regular by the closure properties. – chi Jan 6 '17 at 10:53

To get you started, consider a restricted version of your languge: $$L'=\{ab\mid a\in L_1, b\in L_2\}$$ Notice that $L$ is the concatenation $L'=L_1\circ L_2$. Can you carry on from there?