$7.$ On $=\{a,b\}.$

Draw the DFA figure defined by:

$(q0,a)=\{q0\}$

$(q0,b)=\{q1\}$

$(q1,a)=\{q0\}$

$(q1,b)=\{q2\}$

$(q2,a)=\{q3\}$

$(q2,b)=\{q2\}$

$(q3,a)=\{q0\}$

$(q3,b)=\{q4\}$

$(q4,a)=\{q3\}$

$(q4,b)=\{q4\}$

q $0$ is the initial state and $q_2$ and $q_4$ are the final $($accepting$)$ states

$(12$ points$)$ On $=\{a,b\}.$ Minimize the DFA from question $7($show all your work$)$