$(9$ marks$)$ Based on our discussion of strings and languages in class, list all of the strings in each

of the languages below. A first example is provided for you. See the bottom of the page for some

reminders of definitions, etc.

$$ L $=\{0$

m$1$

n

$|$ m$,$ n in Z

nonneg, m $+$ n $=5\}$

$$ Answer: L $=\{00000,00001,00011,00111,01111,11111\}$

$($a$)$ L$5=\{$w in $\{$a$,$ b$\}$

$$

$||$w$|<=3$ and na$($w$)$ is even$\}$

$($b$)$ L$6=\{$a

mb

nc

p

$|$ m$,$ n$,$ p in Z

nonneg, m $+$ n $+$ p $<=4,$ m $>$ n$,$ n $=$ p$\}$

$($c$)$ L$7=\{$w in $\{$a$,$ b$,$ c$\}$

$$

$||$w$|<=4,$ na$($w$)>$ nb$($w$),$ nb$($w$)=$ nc$($w$)\}$

Reminders:

$$ Z

nonneg is the set of nonnegative integers, $\{0,1,2,3,\}.$

$\{$a$,$ b$\}$

$$

is the set of all possible strings of $$a$$s and $$b$$s$,$ which includes the empty string $.$

$0$ is an even number.

$$ na$($w$)$ is the number of $$a$$s in the string w$,$ nb$($w$)$ is the number of $$b$$s in the string w$,$ etc.

$$ The notation $|$w$|$ represents the length of string w$.$