Exercise $33$ Questions about closure properties of context free languages

$(4$ points$)$

Decide whether the following statements hold for arbitrary languages $L_1,L_2$sube${}^{*}.$ 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:

$x_1=\{a^{2^n}|nin{N}_0\},$

$x_2\frac{?}{b}=ar(x_1),$

$x_3=\{win\{a,b,c{\}}^{*}|$#$a(w)=$#${?}_b(w)=$#${?}_c(w)\}.$

$($a$)$ If $L_2$ is regular and $L_1{L}_2$ is regular, then $L_1$ is context free.

$($b$)$ If the language $(L_1{)}^{*}$ is context free, then $L_1$ is not necessarily also context free.

$($Note: Try to think of a language that is not context$-$free, and that contains $($in addition to

other words$)$ words with length $1.$ Then check if the statement from the task holds true for this

language.$)$

$($c$)$ If $L_1$ is not context free, then $\frac{?}{\text{b}\text{a}\text{r}}(L{)}_1$ is also not context free.

$($d$)$ If $L_1$ is context free and $L_{2}sube{L}_1,$ then $L_2$ is regular.

$($Note: To show that a statement is wrong it suffices to give a counterexample, i$.$e$.$ two languages $L_1,$

$L_2($or one language $L_1)$ for which the statement does not hold.$)$