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Let L be a regular language and M be a context-free language, both over the alphabet Σ. Let Lc and Mc denote the complements of L and M respectively. Which of the following statements about the language Lc∪ Mc is TRUE
- a)It is necessarily regular but not necessarily context-free
- b)It is necessarily context-free.
- c)It is necessarily non-regular.
- d)None of the above
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Let L be a regular language and M be a context-free language, both ove...
Proposition: L is a regular language M is a context free language Derivation: L_c union M_c = complement{L intersection M} Now, L intersection M is a CFL according to closure laws of CFLs, i.e. intersection of a CFL with RL is always a CFL. But, complement{L intersection M} might not be a CFL because complement over CFL doesn't guarantee a CFL. It can even be a RL or it might even lie outside the CFL circle. It will be a context-sensitive language certainly, but nothing else can be said about it. Conclusion: Considering the above derivation, none of the statements are true. Hence correct answer would be (D) None of the above.
Most Upvoted Answer
Let L be a regular language and M be a context-free language, both ove...
Explanation:
To determine the correctness of the given options, we need to analyze the properties of regular and context-free languages and their complements.
Regular Languages:
- A regular language can be recognized by a finite automaton or can be described by a regular expression.
- The complement of a regular language is also regular.
Context-Free Languages:
- A context-free language can be recognized by a pushdown automaton or can be described by a context-free grammar.
- The complement of a context-free language is not necessarily context-free.
Analysis of options:
a) It is necessarily regular but not necessarily context-free.
- The complement of L (Lc) is regular, but we cannot determine the regularity of Mc (complement of M) without additional information. It could be regular or context-free.
- Therefore, option a) is not necessarily true.
b) It is necessarily context-free.
- The complement of L (Lc) is regular, and the complement of a regular language is regular.
- The complement of M (Mc) is not necessarily context-free.
- Therefore, option b) is not necessarily true.
c) It is necessarily non-regular.
- The complement of L (Lc) is regular, and the complement of a regular language is regular.
- The complement of M (Mc) is not necessarily regular.
- Therefore, option c) is not necessarily true.
Conclusion:
None of the given options (a, b, c) are necessarily true. The correct answer is option d) "None of the above."
To determine the correctness of the given options, we need to analyze the properties of regular and context-free languages and their complements.
Regular Languages:
- A regular language can be recognized by a finite automaton or can be described by a regular expression.
- The complement of a regular language is also regular.
Context-Free Languages:
- A context-free language can be recognized by a pushdown automaton or can be described by a context-free grammar.
- The complement of a context-free language is not necessarily context-free.
Analysis of options:
a) It is necessarily regular but not necessarily context-free.
- The complement of L (Lc) is regular, but we cannot determine the regularity of Mc (complement of M) without additional information. It could be regular or context-free.
- Therefore, option a) is not necessarily true.
b) It is necessarily context-free.
- The complement of L (Lc) is regular, and the complement of a regular language is regular.
- The complement of M (Mc) is not necessarily context-free.
- Therefore, option b) is not necessarily true.
c) It is necessarily non-regular.
- The complement of L (Lc) is regular, and the complement of a regular language is regular.
- The complement of M (Mc) is not necessarily regular.
- Therefore, option c) is not necessarily true.
Conclusion:
None of the given options (a, b, c) are necessarily true. The correct answer is option d) "None of the above."
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Let L be a regular language and M be a context-free language, both over the alphabet Σ. Let Lc and Mc denote the complements of L and M respectively. Which of the following statements about the language Lc∪ Mc is TRUEa)It is necessarily regular but not necessarily context-freeb)It is necessarily context-free.c)It is necessarily non-regular.d)None of the aboveCorrect answer is option 'D'. Can you explain this answer?
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Let L be a regular language and M be a context-free language, both over the alphabet Σ. Let Lc and Mc denote the complements of L and M respectively. Which of the following statements about the language Lc∪ Mc is TRUEa)It is necessarily regular but not necessarily context-freeb)It is necessarily context-free.c)It is necessarily non-regular.d)None of the aboveCorrect answer is option 'D'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Let L be a regular language and M be a context-free language, both over the alphabet Σ. Let Lc and Mc denote the complements of L and M respectively. Which of the following statements about the language Lc∪ Mc is TRUEa)It is necessarily regular but not necessarily context-freeb)It is necessarily context-free.c)It is necessarily non-regular.d)None of the aboveCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let L be a regular language and M be a context-free language, both over the alphabet Σ. Let Lc and Mc denote the complements of L and M respectively. Which of the following statements about the language Lc∪ Mc is TRUEa)It is necessarily regular but not necessarily context-freeb)It is necessarily context-free.c)It is necessarily non-regular.d)None of the aboveCorrect answer is option 'D'. Can you explain this answer?.
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