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Prove/Disprove: If $L$ is finite and $R$ is not regular, then $R\cup L$ is not regular.

I think that this one is true, but I am stuck:

Since $R$ is not regular, it is infinite, so $R \cup L$ is also infinite.

Since $R$ is non-regular, it must be that $R \cup L$ is not regular.

It just doesn't feel complete for me...

Any help will be amazing!

Thanks!

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1 Answer 1

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(weird choice of letter $R$ for a non-regular language).

Since $L$ is finite, so is $L\setminus R$ (hence regular).

Now, note that $R = (R\cup L) \setminus (L \setminus R)$. Can you find a contradiction if $R\cup L$ is regular?

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  • $\begingroup$ Very nice! Thank you so much! :) $\endgroup$
    – Mish
    Apr 8 at 22:39

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