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I have this problem that I need to prove that the language is not regular by using pumping lemma, but no matter how much I read how to do it, I still don't understand. Could someone please help how to solve this?

Show that L = { a^n c b^m | n, m are natural numbers and n < m} is not regular.

Welbog
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Saul
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  • Hi and welcome to Stack Overflow. This site works best when you give people the work you've tried so far, so they can help you with specific parts of your solution. With the pumping lemma, are you having trouble finding a pumping string? Or are you having trouble proving that your string can always be pumped into a word outside of the language for any decomposition? I'd suggest pumping the string `a^n c b^(n+1)`, where `n` is arbitrarily large, as required by the pumping lemma. – Welbog Mar 17 '20 at 13:28

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Choose a^p c b^2p. This string is in the language since p < 2p. Pumping any non-empty substring in the first p characters of this string up by a factor of more than p is guaranteed to cause the number of a to increase beyond the number of b. This contradicts the claim of the pumping lemma that doing that on a string in a regular language must give another string in that language. So, the language cannot have been regular.

Patrick87
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