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A regular language L is accepted by a non-deterministic finite automaton (NFA) with n states. Which of the following statement(s) is/are FALSE?

1. L may have an accepting NFA with < n states.
2. L may have an accepting DFA with < n states.
3. There exists a DFA with ≤ 2ⁿ states that accepts L.
4. Every DFA that accepts L has > 2ⁿ states.

Correct answer: Every DFA that accepts L has > 2ⁿ states.

Solution

This statement is false because while a DFA can have up to 2ⁿ states in the worst case when converting from an NFA with n states, it is not guaranteed that every DFA accepting the same language must exceed that number of states; some DFAs may have fewer states.

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