Conversion of Epsilon NFA to DFA

Question

on the following problem: Convert the following ε-NFA to DFA and prove if it is equivalent or not with the A2 in the picture

i think that the epsilon closure of the {3,8} should be {1,2,4,7,8} and i don't know if i made it correct here

Answer 1

I looked carefully at your conversion and it looks good. Now it is easy to see that $\mathcal{A}_1$ and $\mathcal{A}_2$ are not equivalent. For instance, the word $bc^2$ is accepted by $\mathcal{A}_2$ but not by $\mathcal{A}_1$.

For the second part of your question,

I think that the epsilon closure of $\{3,8\}$ should be $\{1,2,4,7,8\}$,

your intuition is wrong, but your computation of the epsilon closure in the table is clear and correct. The reason is that an $\varepsilon$-transition $p \xrightarrow{\varepsilon} q$ cannot be reversed: in other words, the $\varepsilon$-transition allows you to freely go from $p$ to $q$, but not from $q$ to $p$.