$1.[5$ marks$]$ Let $\backslash$Sigma $=\{$a$,$ b$\}.$ Consider the following languages LA and LB over $\backslash$Sigma :

LA $=\{$w in $\backslash$Sigma $|$ w contains an even number of occurrences of symbol a$\}$

LB $=\{$w in $\backslash$Sigma $|$ each pair of consecutive b$$s in w is separated by a substring of a$$s of length $2$i$,$ i $>=0\}$

Clarification: LB is saying that between b$$s there must be $2$i a$$s for some i $>=0($e$.$g$.$ baab in LB

and baaab $/$ in LB $).$ Also, i must be an integer value $>=0.$

For each of the following strings, decide whether it is a member of LA or LB :

$($a$)$ b

$($b$)$ aabbbbaa

$($c$)$ abaaabaaa

$($d$)$ bbaaaabbaab

$($e$)$ a

$2.[5$ marks$]$ Consider the following DFA M :

Describe M as a $5-$tuple M $=($Q$,\backslash$Sigma $,\backslash$delta $,$ q$0,$ F $).$ In particular, give

$($a$)$ Q

$($b$)\backslash$Sigma

$($c$)\backslash$delta $($in the form of a transition table$)$

$($d$)$ q$0$

$($e$)$ F