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Problem 3

# The formal description of a DFA $$M$$ is $\left(\left\\{q_{1}, q_{2}, q_{3}, q_{4}, q_{5}\right\\},\\{\mathrm{u}, \mathrm{d}\\}, \delta, q_{3},\left\\{q_{3}\right\\}\right)$$, where$$\delta$ is given by the following table. Give the state diagram of this machine. $$\begin{array}{c|cc} & \mathrm{u} & \mathrm{d} \\ \hline q_{1} & q_{1} & q_{2} \\ q_{2} & q_{1} & q_{3} \\ q_{3} & q_{2} & q_{4} \\ q_{4} & q_{3} & q_{5} \\ q_{5} & q_{4} & q_{5} \end{array}$$

Expert verified
The state diagram for the given DFA M is as follows: 1. States: $$q_1$$, $$q_2$$, $$q_3$$, $$q_4$$, and $$q_5$$ 2. Input symbols: $$u$$ and $$d$$ 3. Initial state: $$q_3$$ (with an incoming arrow) 4. Final state: $$q_3$$ (double circle) 5. Transitions: $$q_1 \xrightarrow{u} q_1$$, $$q_1 \xrightarrow{d} q_2$$, $$q_2 \xrightarrow{u} q_1$$, $$q_2 \xrightarrow{d} q_3$$, $$q_3 \xrightarrow{u} q_2$$, $$q_3 \xrightarrow{d} q_4$$, $$q_4 \xrightarrow{u} q_3$$, $$q_4 \xrightarrow{d} q_5$$, $$q_5 \xrightarrow{u} q_4$$, and $$q_5 \xrightarrow{d} q_5$$. ![State Diagram](https://i.imgur.com/CrHKSvM.png)
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## Step 2: Draw the State Diagram

Now that we have all the necessary information, we can draw the state diagram for this DFA. 1. Draw a circle for each state, labeling them as $$q_1$$, $$q_2$$, $$q_3$$, $$q_4$$, and $$q_5$$. 2. Mark the initial state, which is $$q_3$$, with an arrow pointing to it. 3. Mark the final state(s), which is $$\{q_3\}$$, by enclosing the state with a double circle. 4. For each entry in the transition table, draw arrows between the states according to the transitions. 5. Label the arrows with the corresponding input symbols. Here is the State Diagram: ![State Diagram](https://i.imgur.com/CrHKSvM.png) Thus, the state diagram for the given DFA is created successfully.

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