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In state diagram of DFA is it mandatory to keep a final state. I mean whether DFA is possible without final state or not?enter image description here

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The definition of DFAs calls for a set of final (aka accepting) states. Nothing requires that set to be non-empty. But note that a DFA without an accepting states can't accept any input, so all such DFAs are essentially equivalent to the one you drew.

Having said that, the class of languages accepted by DFAs wouldn't change if you insisted that every DFA must have at least one accepting state. You could still accept the empty language with an automaton whose state graph was disconnected, with all accepting states unreachable from the initial state. If you insisted on at least one accepting state and a connected state graph, the empty language wouldn't be accepted by any DFA.

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