$1.$This question develops your ability to devise regular expressions, given an explicit definition

of a language. For each of the following languages, prove they are regular by giving a regular expression

which describes them. Justify your answers.

$($a$)$ L $=\{$x $|$ x begins with one $1$ and ends with two $0$s$\}.$

$($b$)$ L $=\{$x $|$ x contains at least three $0$s$\}.$

$($c$)$ L $=\{1,111,$ Sigma$\}.$

$($d$)$ L $=\{$x $|$ the length of x is at most $5\}.$

$($e$)$ L $=\{$x $||$x$|>0,$ i$.$e$.$ x is non$-$empty$\}.$