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Let $L$ be the language generated by regular expression $0^*10^*$ and accepted by the deterministic finite automata $M$. Consider the relation $R_M$ defined by $M$. As all states are reachable from the start state, $R_M$ has ________ equivalence classes.

My Try:

If we draw the DFA for this language then it will have $3$ state, where one is final state and two are non-final states. Then answer should be $3$.

But, official answer is given $6$.

Can you please explain ?

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    Did you draw a totally defined DFA? Recall that $R_{M}$ is defined on $\Sigma^{*} \times \Sigma^{*}$, where $\Sigma$ is your alphabet. – 2017-01-03
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    @ml0105, totally defined DFA? I heard it first time. I newbie with that. – 2017-01-03
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    Frequently, we draw FSMs where the edges only handle the transitions we need to accept the language $L$, and omit the others. In reality, the transition function $\delta : Q \times \Sigma \to Q$ is a total function. So for each (state, letter) pair, you should have a directed edge in your DFA. This is what I mean by a totally defined DFA. – 2017-01-03
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    @ml0105: Yes, it’s fully defined with three states. – 2017-01-03
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    @Mithlesh: Is $R_M$ the relation defined by saying that $x\mathrel{R_M}y$ iff $\hat\delta(s,x)=\hat\delta(s,y)$, where $s$ is the initial state, and $\hat\delta$ is the extended transition function? If so, I agree with you that it has $3$ classes. – 2017-01-03
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    @MithleshUpadhyay: If the answer is $6$, it leads me to believe there was a DFA included with the problem statement. Additionally, the question states: "...and accepted by the deterministic finite automata $M$," which suggests a DFA $M$ was included with the problem. If not, I'd consult with your instructor. – 2017-01-03
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    @ml0105, it's an exam question (please, see qu. no.-55 on page no.- 55): https://drive.google.com/open?id=0B3FDtVFmzdt3bTdGSFQ4SEt3NzQ – 2017-01-03
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    @ml0105, your logic may be correct. $M$ can be another DFA. – 2017-01-03
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    @BrianM.Scott, can you explain given relation, little bit? Please. I know extended transition function. – 2017-01-03
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    @Mithlesh: The relation that I have in mind is the one that says that words $x$ and $y$ are related if they take $M$ to the same state when it starts in the initial state. – 2017-01-03
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    Such an exam question would be closed on this site as "unclear what you are asking"... – 2017-01-03

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