Closure properties of non-context-free languages (concatenation & complement) I am trying to proof the properties of the complement and concatenation of two non-context-free languages $L_1$ and $L_2$.
I believe that both of these languages are closed under complement and concatenation but can't seem to find a solid proof for it. I'm leaning towards them being closed because I can't find any counter examples.
I already proved that non-context-free languages are not closed under union and intersection, so I can use those properties (if applicable).   

So in short what I want to proof, given $L_1, L_2$ are non-context-free languages:
$L_1L_2 \in S_{nonContextFree}$
$\overline{L_1} \in S_{nonContextFree}$
As for closure under complementation -- consider the following hint: context-free languages are not closed under complementation.

Regarding concatenation, this is slightly more tricky, but non-context free languages are not closed under complementation. 

There are many ways of finding counterexamples for the latter, one is the following "trick":

Let $L$ be some non-context free language over some alphabet $\Sigma$, that does not contain $0,1$. Now define two languages: $A=(1\cdot \Sigma^*)\cup (0\cdot L)\cup \{\epsilon\}$ and 
$B=(0\cdot \Sigma^*)\cup (1\cdot L)\cup \{\epsilon\}$

It is not hard to prove that neither $A$ nor $B$ are context free, basically due to presence of $L$.

However, we have that $A\cdot B=\{0,1\}\cdot \Sigma^*\cup \{\epsilon\}$, which is regular (and in particular context-free).