Find a finite automaton such that the language accepted by it is the set of even numbers in binary notation

Question

Use automata theory to prove the following question

Answer 1

Let $L = \{(0+1)^*0\}$. This langage is precisely the set of even integers written in binary notation.

Now just make the following two states NFA :

• $q_0$ is the first state

• $q_1$ is the second states and it's a final state

• $q_0 \overset{\Sigma}{\longrightarrow} q_0$

• $q_0 \overset{0}{\longrightarrow} q_1$

You can also make a three states DFA that solves the problem :

• $q_0$ is the first state
• $q_1$ is the second states and it's final state
• $q_2$ is the third states
• $q_0 \overset{0}{\longrightarrow} q_1$
• $q_0 \overset{1}{\longrightarrow} q_2$
• $q_1 \overset{0}{\longrightarrow} q_1$
• $q_1 \overset{1}{\longrightarrow} q_2$
• $q_2 \overset{0}{\longrightarrow} q_1$
• $q_2 \overset{1}{\longrightarrow} q_2$