Answer
there is no finite-state automaton with two states that recognizes the set of all bit strings that have one or more 1 bits and end with a 0.
Work Step by Step
Showing that there is no finite-state automaton with two states that recognizes the set of all bit strings that have one or more 1 bits and end with a 0.
Suppose that such a machine exists, with start state $s_0$ and other state $s_1$.
- Because the empty string is not in the language but some strings are accepted,
-we must have $s_1$ as the only final state,
- with at least one transition from $s_0$ to $s_1$.
-Because the string 0 is not in the language,
-the transition from $s_0$ on input 0 must be to itself,
-so the transition from $s_0$ on input 1 must be to $s_1$.
-But this contradicts the fact that 1 is not in the language.
