Understanding NFAs and DFAs:

NFA (Non-deterministic Finite Automaton):

An NFA with $ n $ states can have multiple transitions for the same symbol and can include ε-transitions.

DFA (Deterministic Finite Automaton):

A DFA has exactly one transition for each symbol in the alphabet from every state.

Subset Construction:

Converts an NFA to an equivalent DFA by creating states that represent subsets of NFA states.

The resulting DFA may have up to $ 2^n $ states, corresponding to all possible subsets of the NFA's states.

Analyzing Each Option:

Option A: $ k \geq 2^n $

Analysis:

The subset construction can produce up to $ 2^n $ states, but the minimal DFA may have fewer states.

Counterexample:

Consider an NFA that recognizes a language accepted by a DFA with fewer than $ 2^n $ states.

For example, an NFA with $ n $ states that accepts a simple language like $ a^* $ can have a minimal DFA with just one state.

Conclusion:

Not necessarily true.

Option B: $ k \geq n $

Analysis:

The minimal DFA can have fewer states than the NFA.

Counterexample:

An NFA with $ n $ states that is overkill for a simple language.

For instance, an NFA with multiple redundant states accepting a basic language may have a minimal DFA with fewer states.

Conclusion:

Not necessarily true.

Option C: $ k \leq n^2 $

Analysis:

While $ n^2 $ is less than $ 2^n $ for $ n > 4 $, the subset construction can produce a DFA with up to $ 2^n $ states, which can be greater than $ n^2 $.

Counterexample:

For larger $ n $, $ 2^n $ exceeds $ n^2 $, so the minimal DFA may have more than $ n^2 $ states.

Conclusion:

Not necessarily true.

Option D: $ k \leq 2^n $

Analysis:

The subset construction method guarantees that the equivalent DFA will have at most $ 2^n $ states.

The minimal DFA cannot have more states than the DFA produced by the subset construction.

Reasoning:

Every state in the DFA corresponds to a subset of NFA states.

Therefore, the total number of possible subsets (and hence DFA states) is $ 2^n $.

Conclusion:

Necessarily true.

Final Answer:

Only Option D is necessarily true for any NFA with $ n $ states:

$ k \leq 2^n $

Answer: Option D