https://r2.erweima.ai/mathbot/image/1724431544131z0otoqli.jpg

https://r2.erweima.ai/mathbot/image/1724431544132se18shba.jpg

Let's analyze and solve the two questions related to the automaton depicted in the images you've uploaded.

### Question 3: Determine which strings are accepted by the automaton.
The automaton is a Deterministic Finite Automaton (DFA). The task is to determine which of the given strings are accepted by the DFA.

**Steps to solve:**
1. **Initial State:** Start at state \( q_0 \).
2. **Transition Through States:** Follow the transitions according to the input string.
3. **Final State:** Check if the final state after consuming the entire string is an accepting state (states with double circles).

**Strings to verify:**
- \( a) \) 0110
- \( b) \) 1001
- \( c) \) 1110
- \( d) \) 10011

#### String \( a) \) 0110
- \( q_0 \xrightarrow{0} q_1 \)
- \( q_1 \xrightarrow{1} q_1 \)
- \( q_1 \xrightarrow{1} q_1 \)
- \( q_1 \xrightarrow{0} q_2 \) (final state: \( q_2 \), not accepting)

**Result:** **Not Accepted**

#### String \( b) \) 1001
- \( q_0 \xrightarrow{1} q_4 \)
- \( q_4 \xrightarrow{0} q_5 \) (final state: \( q_5 \), accepting)

**Result:** **Accepted**

#### String \( c) \) 1110
- \( q_0 \xrightarrow{1} q_4 \)
- \( q_4 \xrightarrow{1} q_4 \)
- \( q_4 \xrightarrow{1} q_4 \)
- \( q_4 \xrightarrow{0} q_5 \) (final state: \( q_5 \), accepting)

**Result:** **Accepted**

#### String \( d) \) 10011
- \( q_0 \xrightarrow{1} q_4 \)
- \( q_4 \xrightarrow{0} q_5 \)
- \( q_5 \xrightarrow{0} q_5 \)
- \( q_5 \xrightarrow{1} q_3 \) (final state: \( q_3 \), accepting)
- \( q_3 \xrightarrow{1} q_3 \) (final state: \( q_3 \), accepting)

**Result:** **Accepted**

**Final answer for Question 3:** Strings \( b), c), d) \) are accepted by the automaton.

### Question 4: Describe the language accepted by the automaton.
The task here is to describe the language accepted by the automaton in set notation.

**Steps to solve:**
1. **Analyze the transitions:**
   - State \( q_5 \) is an accepting state.
   - From \( q_0 \), any string starting with \( 1 \) followed by any combination of \( 0 \)s and \( 1 \)s is accepted as long as it can reach state \( q_5 \).

2. **State \( q_1 \):**
   - It accepts strings that have a prefix of \( 0 \), then loop on \( 1 \), and finally end with \( 0 \).

**Language Description:**
- The language \( L \) can be described as strings that:
  - Start with \( 1 \), followed by any combination of \( 0 \)s and \( 1 \)s.
  - Alternatively, the strings may start with \( 0 \), followed by one or more \( 1 \)s, and end with a \( 0 \).
  
**In set notation:**  
\[ L = \{ w \in \{0,1\}^* \mid w \text{ ends in 0 or } w \text{ starts with 1} \} \]

**Final answer for Question 4:** The language accepted by the automaton is defined as \( L = \{ w \in \{0,1\}^* \mid w \text{ ends in 0 or } w \text{ starts with 1} \} \).

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Would you like any further details or explanations? 

Here are 5 additional questions to expand on this information:
1. How would the automaton behave with the string "0011"?
2. Can a string with alternating 0s and 1s be accepted by this automaton? Why or why not?
3. What changes would you make to the DFA if it had to reject any string that starts and ends with 1?
4. How can the language description be altered if state \( q_2 \) also becomes an accepting state?
5. What would be the effect of adding a transition from \( q_0 \) to \( q_5 \) on input \( 1 \)?

**Tip:** Understanding DFAs is essential for grasping more complex computational models like Non-Deterministic Finite Automata (NFA) and Turing Machines.

Explore detailed solutions to Deterministic Finite Automaton (DFA) problems, including analyzing acceptance of strings and describing accepted languages. This page covers key concepts in automaton theory suitable for advanced undergraduate studies.

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