Section $2\mathrm{.}1$ Deterministic Finite Acceptors $($DFA$)$

$(3$ points$)$ Section $2\mathrm{.}1$ Questions $3$ and $4$

For $=\{a,b\},$ construct dfa's that accept the sets consisting of

I. $**$all strings with and even number of $a\text{'}$s

II$.$ All strings with an even number of $a\text{'}$s and an odd number of $b\text{'}$s

III. $**$all strings with exactly one $a$

IV$.$ All strings with at least one $b$ and exactly $2a\text{'}$s

$(3$ points$)$Section $2\mathrm{.}1$ Question $7$: $($The answer to III is at the bottom of this document$)$

Find dfa's for the following languages on $=\{a,b\}.$

I. $,****L=\{w$:$|w|mod30\}$

II$.L=\{w$:$n_b(w)mod3>1\}$

III. $L=\{w$:$|w|mod5=0,|w|5\}\{aba{b}^n$:$n0\}\{ab{a}^n$:$n0\}L=\{a^n$:$n0\}\{b^{n}a$:$n1\}$