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For the following language, write grammar independent of the text.

$$\{a^i b^j: j < 2i \text{ and } j \ne i \} $$

I want a hint to start solving this problem. Where should I start?

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If $j \neq i$ then either $j < i$ or $j > i$. Hence you can write your language as $$ \{ a^i b^j : j < i \} \cup \{ a^i b^j : i < j < 2i \}. $$ It suffices to find a grammar for each of these two languages. Such questions have been asked and answered on this site.

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  • $\begingroup$ This site isn’t for doing your homework for you. $\endgroup$ – Yuval Filmus Apr 20 at 14:18
  • $\begingroup$ You are right :) $\endgroup$ – hermi Apr 20 at 14:37

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