Question $1$: $(20$ pts$)$

Given the language L$1$ over the alphabet $\{a,b\},$ where L$1$ consists of all strings that contain

an even number of a$\text{'}$s$,$ construct a regular expression that describes L$1.$

Question $2$: $(20$ pts$)$

Design a DFA for the language L$2$ over the alphabet $\{a,b\},$ where L$2$ consists of all strings

that end with the substring $"$ab$".$

Question $3$: $(20$ pts$)$

Describe the language accepted by this DFA.

Question $4$: $(20$ pts$)$

Given the regular expression $(a+b{)}^{**}$bab$(a+b{)}^{**},$ describe the language generated by this

regular expression.

Question $5$: $(10$ pts$)$

Consider the DFA from Question $3.$ Determine whether the strings "abaaa" and "baaab" is

accepted by this DFA. Justify your answer by showing the transitions.

Question $6$: $(10$ pts$)$

Given the regular expression a$,$ verify if the strings "abba" and "aaabbaa" can be

generated by this regular expression. Show the steps.