1. **Problem Statement:** Identify the language recognized by the given DFA with states 0 to 7, initial state 0, and terminal states 5, 6, and 7. 2. **Understanding the DFA:** The DFA transitions are: - From 0: 0\to1, 1\to2 - From 1: 0\to3, 1\to4 - From 2: 0\to4 - From 3: 0\to6, 1\to5 - From 4: 0\to5 - From 5: 0 or 1\to7 - From 6: 0 or 1\to6 (loop) - 7 is terminal with no outgoing edges. 3. **Terminal states:** 5, 6, 7 4. **Analyzing paths to terminal states:** - To reach 6: must go 0 (from 0 to 1), 0 (1 to 3), 0 (3 to 6), then stay in 6 with any input. - To reach 5: can go 0,1,1 (0 to 1 to 4 to 5) or 0,0,1 (0 to 1 to 3 to 5) or 1,0,0 (0 to 2 to 4 to 5). - To reach 7: from 5 with any input 0 or 1. 5. **Language description:** - Strings starting with 0, then 0, then 0, followed by any combination of 0s and 1s (loop in 6). - Strings that reach 5 by paths described, then optionally followed by 0 or 1 to reach 7. 6. **Formalizing:** - The language includes strings over {0,1} that: - Start with 000 and then any string (for state 6), or - Start with 001 or 011 or 100 and then possibly one more symbol (for states 5 and 7). 7. **Summary:** The language recognized is all strings over {0,1} that lead from 0 to terminal states 5, 6, or 7 following the described transitions. **Final answer:** The language recognized by the DFA consists of all strings over {0,1} that either: - Contain the prefix 000 followed by any string (looping in state 6), or - Contain prefixes 001, 011, or 100 leading to state 5, optionally followed by one more symbol to reach state 7.