$(20$pt$)$ Consider the switch statement in C$-$like languages, defined here according to this grammar:

Switch $\backslash$longrightarrow switch $($ expr $)\{$ Cases Default $\}$

Cases $\}\backslash$longrightarrow$\backslash$mathrm$\{$ Cases Case

Cases $\}\backslash$longrightarrow

Case $\}\backslash$longrightarrow$\backslash$mathrm$\{$ case const: Stmt

Stmt $\backslash$longrightarrow stmt; break;

Stmt $\}\backslash$longrightarrow$\backslash$mathrm$\{$ stmt;

Stmt $\backslash$longrightarrow $\}$

Default $\backslash$longrightarrow default: stmt;

Default $\backslash$longrightarrow

$($a$)(2$pt$)$ Compute the sets FIRSt$(x)$ and FOLlOW$(x),$ for all non$-$terminals $x,$ and the sets

PREDICT$(p)$ for all production rules $p.$

$($b$)(3$pt$)$ Prove that $G$ is not an $LL(1)$ grammar. Describe all conflicts.

$($c$)(5$pt$)$ Build an LL$(1)$ grammar, $G_1,$ equivalent with $G.$ Compute, for $G_1,$ the sets FIRST $(x)$

and Follow $(x),$ for all non$-$terminals $x,$ and the sets PrEdiCT$(p)$ for all production rules $p.$

$($d$)(5$pt$)$ Show how the switch statement can be used, in general, to simulate any if $..$then $..$else

statement.

$($e$)(5$pt$)$ How is it possible that the switch statement can simulate if's and have an LL$(1)$

grammar, since we said that there is no top$-$down grammar for if $..$then. $.$ else statements?