Language accepted by given NFA

Question

Language accepted by following NFA and number of states in DFA accepting that Language are:

1. $\{a^n|n=2k,kϵN\}$ and 2

2. $\{a^{2n}|n=2k,kϵN\}$ and 2

3. $\{a^n|n=2k,kϵ N\}$ and 3

4. $\{a^{2n}|n=2k,kϵ N\}$ and 3 

Comments

The answer is Option 3

Language Accepted here is "Even Number of a's EXCLUDING epsilon" and the

No. Of States in DFA must be 3.

2 States for "Accepting Even Number of String for Single Alphabet={a}" and 1 State for Dead State.

Correct me If I am wrong.

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So the solution of testbook is incorrect, right? I indeed checked option 3, but they are saying option 1!!!

Answer 1

(1) since from the initial state in the given NFA, if one encounters even number of  a's , then and only then reaches final states. For example, from state S , for string "aaaa", one reaches the bottom right-most state. But, as one can notice, it also has an epsilon-transition to S, which is a final state. Hence, we can conclude here that this NFA accepts strings having even number of a's. 

Design of DFA is pretty trivial and hence, left as an exercise for the person who asked this question.

Thank You.

Comments

So the solution of testbook is incorrect, right? I indeed checked option 3, but they are saying option 1!!!

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No.. the testbook solution is correct... I have edited my answer.. Minimum DFA has 2 states

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How Minimum DFA has 2 State?

Here It is Accepting Epsilon Also Isn't It?

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Epsilon means what? 0 right? that means 0 a's. 0 is even or odd?