Decide whether the following statements hold for arbitrary languages L$1,$ L$2\backslash$Sigma

$$

$.$ In each case, give

either a proof or a counterexample. Answers without any motivation achieve no points.

If necessary, you can use that the following languages are not context free:

X$1=\{$a

$2$

n

$|$ n in N$0\},$

X$2=$ X$1,$

X$3=\{$w in $\{$a$,$ b$,$ c$\}$

$$

$|$ #a$($w$)=$ #b$($w$)=$ #c$($w$)\}.$

$($a$)$ If L$2$ is regular and L$1\backslash$cap L$2$ is regular, then L$1$ is context free. $(1$ p$)$