Consider the languages

L$1=$ L$($a $+$ b

$$

a$($ab $+$ ba$)$

$$

a

$$

$),$

L$2=$ L$(($ba$)$

$$

a$($aa$)$

$$

a$),$

L$3=$ L$($a

$$

$($aa $+$ baa$)$a $+$ aaa$).$

a$)$ Give a word that is in all three languages, and a word that is in none of them. $(1$pt$)$

b$)$ Show that each of the languages L$1,$ L$2$ and L$3$ is different. $(1\mathrm{.}5$pt$)$

c$)$ Consider the language L$,$ given by

L$4=\{$w in A

$$

$|$ the word bab occurs at least twice in w$\}$

Is L$4$ equal to L$5=$ L$($a

$$

$($bab$)$a

$$

b

$$

$($ab $+$ bab$))?$ Justify your answer$2$ Regular Expressions $($I$)$

Consider the languages

$L_1=L(a+b^{**}a(ab+ba{)}^{**}{a}^{**}),$

$L_2=L((ba{)}^{**}a(aa{)}^{**}a),$

$L_3=L(a^{**}(aa+$baa$)a+$aaa$).$

a$)$ Give a word that is in all three languages, and a word that is in none of them.

$(1$pt$)$

b$)$ Show that each of the languages $L_1,L_2$ and $L_3$ is different.

c$)$ Consider the language $L,$ given by

$L_4=\{win{A}^{**}|$ the word bab occurs $at$ least twice $inw\}$

Is $L_4$ equal to $L_5=L(a^{**}(bab)a^{**}{b}^{**}(ab+$bab$))?$ Justify your answer