We call the word $x_1$ a true prefix of the word $x$, if a non-empty word $x_2$ exists so that $x=x_1x_2$. For the language L (over some finite $abc$..). We define MAX(L) as:
$MAX(L)$ = {$w_1 \in L $| $w \notin L$ so that $w_1$ is a true prefix of $w$}
prove (or disprove): If the language L is CF then the language $MAX(L)$ is CF.
I think that the correct direction here is to disprove. The question is, how do I disprove it for a general $L$?
I know how to use the pumping lemma to disprove a certain language, but I'm not sure if it can work for a general $L$?
Am I going in the right direction or am I completely wrong?