I have been asked a very interesting question in preparation for my exam coming up. The question is as follows:
Is $L=\{a^n b^n : n \geq 0\} \cap \{a^*b^*\}$ regular or not?
Assume $L$ is regular. Then, $L^c$ should be regular as well. Thus, $L^c = \{a^n b^n : n \lt 0\} = \{\}$, so if I compliment the compliment, I should get $L = U$ (the universal set). So, then $L=\{a^n b^n : n \geq 0\} \cap \{a^*b^*\} \neq U$, because $L$ does not contain words of the form $baba$, or $baaaababa$, etc. Therefore the language is not regular.
Is this an acceptable answer for the upcoming exam? Does this persuade anyone?