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Michael Sipser Edition 3 Exercise 1 Question 39 (Page No. 89)

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The construction in $\text{Theorem 1.54}$ shows that every $\text{GNFA}$ is equivalent to a $\text{GNFA}$ with only two states$.$ We can show that an opposite phenomenon occurs for $\text{DFAs.}$ Prove that for every $k > 1,$ a language $A_{k} ⊆ \{0,1\}^{*}$ exists that is recognized by a $\text{DFA}$ with $k$ states but not by one with only $k − 1$ states$.$
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