I have a language:
$$ \{a^jb^k \mid j \neq k \text{ and } j \equiv k \pmod 3 \}$$
I want to prove that this language is regular. My first thought was to create a regular expression that accounts for all three of the cases of modulo 3 like such:
$$(aaa)^*(bbb)^* | a(aaa)^*b(bbb)^* | aa(aaa)^*bb(bbb)^*$$
However, this doesn't account for the fact that $j$ cannot equal $k$. Is there a way to exclude certain cases like this in a regular expression?