Theorem : For any regular language L there is a unique (upto isomorphism as defined ) DFA that has a minimum number of states. In fact, the minimum DFA is the same as the one that has as states the equivalence classes of (as defined in the context of Myhill-Nerode Theorem).

Proof : Let be the DFA which states are equivalence classes of . Let be any other DFA recognizing L. we have already shown that

• is a right invariant equivalence relation of finite index s.t. L is the union of some of its equivalence classes.
• is a refinement of .
• This implies, the number of equivalence classes of (which is equal to the number of states in M) must be greater than or equal to the number of equivalence classes of ( which is equal to the number of states in , by construction ).
• That is

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