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I am supposed to minimize the following DFA automa, which contains dead state:

https://imgur.com/F9KkIss

But after the minimization of the automata, it stayed the same.

Is it correct?

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I didn't try to minimize your particular automaton but it is definitely possible for a minimal DFA to have a dead state. Keep in mind that often such states are omitted from the graphical representation of the automaton (nevertheless they are still part of the set of states of the DFA).

Think, for example, of a minimal DFA that recognizes $L=\{a\}$ over the alphabet $\Sigma = \{a, b\}$.

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