Determine whether a given language $L$ is regular , CFL or nither. Let > $$L=\left\\{w\in\\{a,b,c\\}^{\ast}\Bigg\vert \exists \sigma_1,\sigma_2\in\\{a,b,c\\}\text{ s.t } \\#_{\sigma_1}(w)\ne \\#_{\sigma_2}(w)\right\\}$$ Determine whether $L$ is regular, context free or nither. It is clear to me that $L$ can't be regular because we can't count the number of $\sigma_1$ and $\sigma_2$. I believe it is context free, but I couldn't manage to construct proper PDA or a CFG in order to show it. Please help, thanks.

A word is in $L$ iff either the number of $a$'s is not equal to the number of $b$'s, or the number of $c$'s, or if the number of $b$'s and $c$'s is unequal, hence we have \begin{align*} L = & \\{ w \in \\{a,b,c\\}^{\ast} \mid \\#_a(w) \
e \\#_b(w) \\} \\\ & ~ \cup \\{ w \in \\{a,b,c\\}^{\ast} \mid \\#_a(w) \
e \\#_c(w) \\} \\\ & ~ \cup\\{ w \in \\{a,b,c\\}^{\ast} \mid \\#_b(w) \
e \\#_c(w) \\}. \end{align*} As the context-free languages are closed under union, you just have to find a grammar for $\\{ w \in \\{a,b,c\\}^{\ast} \mid \\#_a(w) \
e \\#_b(w) \\}$ and similar the other languages.

Do you know how to do that? Maybe you should think about a CFG for $$ \\{ w \in \\{a,b\\}^{\ast} \mid \\#_a(w) \
e \\#_b(w) \\} $$ first and then generalise!