I'm currently working on some homework dealing with removing epsilon transitions and I wanted to quickly run 2 pieces of work by you guys to confirm if I have the right idea here.
Above is the NFA that I am working with, I have created the transition table which contains my a,b,ECLOSURE,a* and b* transition, I don't want to post the whole table because this is graded and I do not want to violate the academic rules, instead I will only post a small section.
State 1:
$a \rightarrow \emptyset$
$b \rightarrow \emptyset$
$\epsilon \rightarrow \{2,3\}$
$ECLOSURE \rightarrow \{1,2,3,4,5\}$
$a^* \rightarrow \{4,7,8\}$
$b^* \rightarrow \{3,5,6,8\}$
Sorry about the formatting, I can't seem to get a handle on the formatting here.
My logic for $a^*$ is that the a-transtions of $\{1,2,3,4,5\}$ result in $\{4,7,8\}$ since $1$ goes to $\emptyset$, $2$ goes to $4$ and $3$,$4$ and $5$ all transition to $7$. Following that, we run the ECLOSE of $4$ and $7$ which transition to $4$ and $8$ respectively, hence the $\{4,7,8\}$.
I followed the same logic for the $b^*$.
I am not looking for someone to do the question for me, I would like someone to point out any mistakes in my logic, I have tried this question twice now and both times I received different outcomes for my transition table. I believe that I finally have it down but I just wanted some confirmation, as there are 3 more parts to this assignment which all rely on the result of the current question I am working on.
Thank you for your time.
edit: removed the state $4$ from $b^*$, I read wrong and realized it shouldn't be in $b^*$
Furthermore, I wanted to add in the $a^*$ transition for state 2 just to have an extra example of my work,
$a^* \rightarrow \{4,7,8\}$
I used the same logic as above to construct this transition set.