$1.$ For any string w $=$ w$1$w$2$ wn the reverse of w$,$ written wR$,$ is the string w in reverse order, wn $$ w$2$w$1.$ For any language A$,$ let AR $=\{$wR $|$ w in A$\}.$ Show that if A is regular then so is AR$.$ In other words, prove that regular languages are closed under the operation of string reversal.

$2.$ Let L$1,$ L$2,$ L$3$ be languages over the alphabet $\{$a$,$ b$,$ c$,$ d$\}$ and let L denote the

complement of a language L$.$ If

$$ L$1$ L$2=$ L$3$ and

$$ L$3$ is not regular,

then what, if anything, can you say about the languages L$1$ and L$2?$ Explain your answer.

$3.[3$ marks$]$ Provide regular expressions for the following sets given the alphabet $=\{$a$,$ b$,$ c$\}$:

$($a$)\{$w $|$ every odd position of w is c$\}$

$($b$)\{$w $|$ w does not contain the substring bc$\}$

$($c$)\{$w $|$ w contains at least two b$$s$,$ at most one c$,$ and no a$$s$\}$