When you convert epsilon NFA to NFA, how do you decide the final states of resultant NFA?

Question

The question is-:

THIS is the transition table for NFA-:

Final result as shown in youtube video.

https://www.youtube.com/watch?v=GjLiXk0imi0&list=PLENQMW_c1dimRCKF3bjUqHaH8dvJkapSw&index=49

My question-: What are the final states of this nfa?

I think q0, q1 and q2 all are final states. Because $\in$ closure of q0,q1 gives q2. And q2 is the final state itself. So q2 U (q1 U q0) is the final state.

Another youtuber solves the same problem but gets different answer-: (exactly what I am telling and tbh this makes more sense than the first one)

Source-: https://www.youtube.com/watch?v=ce0xAcABOYw

Then this neso academy video does the same (as I said).

https://www.youtube.com/watch?v=WSGcmaHNBFM

Answer 1

In short your assumption: "I think q0, q1 and q2 all are final states." is true.

Suppose your $\epsilon-\text{NFA}$ is called $M_1$ and its equivalent $\text{NFA}$ is called $M_2$. every state in $M_1$ that can see at least one final state by only getting input $\epsilon$, will be a final state in $\text{NFA}$.

I think the reason will be quite obvious if you would consider $q_1$ in your own example of $\epsilon-\text{NFA}$.
A final state, $q2$, is in its $\epsilon^{*}$, i.e, being in state $q_1$ (after starting from the initial state and getting the input $w$) can end in $q_2$ and the given input will be accepted by $M_1$.
So in $M_2$ the state $q_1$ is a final state too (as long as $q_0$ with the same proof).

Answer 2

In the powerset construction $2^{\mathcal{A}}$ of an automaton $\mathcal{A}$, the final states of $2^{\mathcal{A}}$ are the sets that contain a final state of $\mathcal{A}$.

This also works when removing $\varepsilon$-transitions.