Sol.

Step 01: Find ∈-closure of (q1), (q2) and (q3).

∈-closure of (q1) = {q1, q2, q3}

∈-closure of (q2) = {q2, q3}

∈-closure of (q3) = {q3}

For each state find the next state for each input. See the table below,

State | 0 | 1 | 2 |

->q1 | {q1,q2,q3} | {q2,q3} | {q3} |

q2 | φ | {q2,q3} | {q3} |

q3 | φ | φ | {q3} |

From the question diagram, it is clear that only with ∈ input q1 and q2 state can reach to the final state.

So, now without ∈ input, q1 and q2 is also treated as final states.

As shown in diagram below.