Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Let L1 be a recursive language, and let L2 be...
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Let L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?
L1' --> Complement of L1
L2' --> Complement of L2
L2' --> Complement of L2
- a)L1' is recursive and L2' is recursively enumerable
- b)L1' is recursive and L2' is not recursively enumerable
- c)L1' and L2' are recursively enumerable
- d)L1' is recursively enumerable and L2' is recursive
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Let L1 be a recursive language, and let L2 be a recursively enumerable...
Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively enumerable.
Recursive Languages are closed under complementation, but recursively enumerable are not closed under complementation. If a languages L is recursively enumerable, then the complement of it is recursively enumerable if and only if L is also recursive. Since L2 is recursively enumerable, but not recursive, L2' is not recursively enumerable.
Recursive Languages are closed under complementation, but recursively enumerable are not closed under complementation. If a languages L is recursively enumerable, then the complement of it is recursively enumerable if and only if L is also recursive. Since L2 is recursively enumerable, but not recursive, L2' is not recursively enumerable.
Most Upvoted Answer
Let L1 be a recursive language, and let L2 be a recursively enumerable...
Explanation:
To understand why option B is the correct answer, we need to understand the definitions of recursive languages and recursively enumerable languages.
Recursive Language:
A language L is recursive if there exists a Turing machine that can decide whether any given input belongs to L or not. In other words, the Turing machine will always halt and give a definite answer of "yes" or "no" for any input.
Recursively Enumerable Language:
A language L is recursively enumerable if there exists a Turing machine that can enumerate (list) all the strings in L. It may not halt or give an answer for strings that are not in L.
Option A:
If L1 is recursive, then its complement (denoted as L1') is also recursive. Similarly, if L2 is recursively enumerable, then its complement (denoted as L2') is also recursively enumerable. Therefore, both L1 and L2 will have the same properties (recursive or recursively enumerable). So, option A is incorrect.
Option C:
If both L1 and L2 are recursively enumerable, then their complements (L1' and L2') will also be recursively enumerable. However, the question states that L2 is not a recursive language. Therefore, option C is incorrect.
Option D:
If L1 is recursively enumerable and L2 is recursive, then their complements (L1' and L2') will also have the same properties (recursively enumerable or recursive). Therefore, option D is incorrect.
Option B:
If L1 is a recursive language, then its complement (L1') will not be recursively enumerable, as recursive languages are closed under complementation. Similarly, if L2 is recursively enumerable, then its complement (L2') will not be recursive, as recursively enumerable languages are not closed under complementation. Therefore, option B is correct.
Conclusion:
The correct answer is option B: L1 is recursive and L2 is not recursively enumerable.
To understand why option B is the correct answer, we need to understand the definitions of recursive languages and recursively enumerable languages.
Recursive Language:
A language L is recursive if there exists a Turing machine that can decide whether any given input belongs to L or not. In other words, the Turing machine will always halt and give a definite answer of "yes" or "no" for any input.
Recursively Enumerable Language:
A language L is recursively enumerable if there exists a Turing machine that can enumerate (list) all the strings in L. It may not halt or give an answer for strings that are not in L.
Option A:
If L1 is recursive, then its complement (denoted as L1') is also recursive. Similarly, if L2 is recursively enumerable, then its complement (denoted as L2') is also recursively enumerable. Therefore, both L1 and L2 will have the same properties (recursive or recursively enumerable). So, option A is incorrect.
Option C:
If both L1 and L2 are recursively enumerable, then their complements (L1' and L2') will also be recursively enumerable. However, the question states that L2 is not a recursive language. Therefore, option C is incorrect.
Option D:
If L1 is recursively enumerable and L2 is recursive, then their complements (L1' and L2') will also have the same properties (recursively enumerable or recursive). Therefore, option D is incorrect.
Option B:
If L1 is a recursive language, then its complement (L1') will not be recursively enumerable, as recursive languages are closed under complementation. Similarly, if L2 is recursively enumerable, then its complement (L2') will not be recursive, as recursively enumerable languages are not closed under complementation. Therefore, option B is correct.
Conclusion:
The correct answer is option B: L1 is recursive and L2 is not recursively enumerable.
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Let L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?L1' --> Complement of L1L2' --> Complement of L2a)L1' is recursive and L2' is recursively enumerableb)L1' is recursive and L2' is not recursively enumerablec)L1' and L2' are recursively enumerabled)L1' is recursively enumerable and L2' is recursiveCorrect answer is option 'B'. Can you explain this answer?
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Let L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?L1' --> Complement of L1L2' --> Complement of L2a)L1' is recursive and L2' is recursively enumerableb)L1' is recursive and L2' is not recursively enumerablec)L1' and L2' are recursively enumerabled)L1' is recursively enumerable and L2' is recursiveCorrect answer is option 'B'. 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 L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?L1' --> Complement of L1L2' --> Complement of L2a)L1' is recursive and L2' is recursively enumerableb)L1' is recursive and L2' is not recursively enumerablec)L1' and L2' are recursively enumerabled)L1' is recursively enumerable and L2' is recursiveCorrect answer is option 'B'. 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 L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?L1' --> Complement of L1L2' --> Complement of L2a)L1' is recursive and L2' is recursively enumerableb)L1' is recursive and L2' is not recursively enumerablec)L1' and L2' are recursively enumerabled)L1' is recursively enumerable and L2' is recursiveCorrect answer is option 'B'. Can you explain this answer?.
Let L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?L1' --> Complement of L1L2' --> Complement of L2a)L1' is recursive and L2' is recursively enumerableb)L1' is recursive and L2' is not recursively enumerablec)L1' and L2' are recursively enumerabled)L1' is recursively enumerable and L2' is recursiveCorrect answer is option 'B'. 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 L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?L1' --> Complement of L1L2' --> Complement of L2a)L1' is recursive and L2' is recursively enumerableb)L1' is recursive and L2' is not recursively enumerablec)L1' and L2' are recursively enumerabled)L1' is recursively enumerable and L2' is recursiveCorrect answer is option 'B'. 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 L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?L1' --> Complement of L1L2' --> Complement of L2a)L1' is recursive and L2' is recursively enumerableb)L1' is recursive and L2' is not recursively enumerablec)L1' and L2' are recursively enumerabled)L1' is recursively enumerable and L2' is recursiveCorrect answer is option 'B'. Can you explain this answer?.
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