Consider FA $(1)$ and FA $(2)$ at end of this question. Let

L$1$ be the language accepted by FA $(1)$ and let L$2$ be

the language accepted by FA $(2).$

$($a$)$ Using the algorithm of Kleene's theorem, Lemma $3,$

Rule $2,$ construct an FA for the union language L$1+$ L$2.$

Show your work. $(10$ Points$)$

$($b$)$ Give an example of a word in the language L$1+$L$2$

that is also in both languages L$1$ and L$2.(02$ Points$)$

$($c$)$ Give a word in the language L$1+$ L$2$ that is also in L$1,$

but not in L$2.(02$ Points$)$

$($d$)$ Give a word in the language L$1+$ L$2$ that is also in L$2,$

but not in L$1.(02$ Points$)$