For each of the following languages, give a context$-$free grammar that generates the language. Brie$$y justify why your

grammar generates that language. In addition, for part a$,$ write a detailed proof of your claim.

$($a$)$ f$0$

i$1$

i

ji $0$g

$($b$)$ f$0$

i$1$

j$0$

k

ji $=$ j $\_$ j $=$ kg

$($c$)$ f$0$

i$1$

j

ji j $2$ig

$($d$)$ fw $2$ f$0$; $1$g

j#$1($w$)=$ #$0($w$)$g

$($e$)$ fx $2$ f$0$; $1$g

j:$((9$i; j$)($x $=(0$i$1$

i

$)$

j

$))$g