$\{a^n b^n\} \cap \{a^*b^*\}$ regular or not?
The complement of $\{a^nb^n \mid n \geq 0\}$ is not $\{a^nb^n \mid n &lt; 0\}$. And the complement of the complement should always equal the original set. The rest of your argument falls apart here.

As an example, abb is a string that is not $a^n b^n$, thus it should be in the complement of L.



To hint at an answer, is there any string $a^nb^n$ that is not $a^*b^*$? What does this say about the intersection of the two?