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Find the equivalence classes of a nerode theorem and use equivalence classes to construct a reduced DFA for the following language: $π‘Ž^+(𝑏+πœ€)𝑐^*$

The answer: $πœ€,π‘Ž^+,π‘Ž^+𝑏𝑐^βˆ—+π‘Ž^+𝑐^+,(𝑏+𝑐)Ξ£^βˆ—+𝐿⋅(π‘Ž+𝑏)Ξ£^βˆ—$

I figured that the minimal base for the language is $a^+$ and for the $π‘Ž^+𝑏𝑐^βˆ—+π‘Ž^+𝑐^+$ its just multiplying the expression with $(𝑏+πœ€)$ but I didn't figure where the $c^+$ came from. Isn't $c^* πœ€ = c^*?$

Now for the set and $(𝑏+𝑐)Ξ£^βˆ—+𝐿⋅(π‘Ž+𝑏)Ξ£^βˆ—$ I don't know what's the intuition for that and how they came up with it.

If im not wrong, epsilon is an equivalent set because it's competent of L

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  • $\begingroup$ What is your question? We require you to articulate a specific question about your situation. I am having a hard time determining what exactly your question is. $\endgroup$
    – D.W.
    Commented Jul 18, 2023 at 17:30

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