going from equivalnce classes into a DFA

Question

Hello all,

I saw an interesting question and i was wondering how to solve it. basically the subject i am having trouble with is going from equivalence classes of $R_L$ to building a DFA.

The question:

L is a language over {0,1}, for which the equivalence classes of $R_L$ are:

$ \{w|\#_0(w) \is\ even\ and\ \#_1(w) \ is\ even\}$

$ \{w|\#_0(w) \is\ even\ and\ \#_1(w) \ is\ odd\}$

$ \{w|\#_0(w) \is\ odd\ and\ \#_1(w) \ is\ odd\}$

$ \{w|\#_0(w) \is\ odd\ and\ \#_1(w) \ is\ even\}$

additional information: $\epsilon \in L$ and $0, 1, 1110 \not \in L$

how do you go from equivalence classes to finding the DFA? don’t understand it.

thank you very much for your help

Comments

it is dfa over input alphabet 0 and 1 where no of 0’s are even and no of 1’s are even

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*please note this is an equivalence class so empty string should be present in only one partition (even,even) , so two accepting states will be in 1st class only . Here is a draft i came up with , while satisfying the restrictions , do verify!

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thank you so much!