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If L1 and L2 are a pair of complementary languages. Which of the following statement is not possible?
- a)Both L1, and L2 are recursive.
- b)L1 is recursive and L2 is recursively enumerable but not a recursive.
- c)Neither L1, nor L2 is recursively enumerable.
- d)One is recursively enumerable but not recursive, the other is is not recursively enumberable.
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If L1 and L2 are a pair of complementary languages. Which of the follo...
The correct statement follows from following theorem:
Theorem: If L1 and L2 is a pair of complementary language then either
• Both L1 and L2 are recursive.
• Neither L1, and L2 are recursively enumerable.
• One is recursively enumerable but not recursive, the other is not RE.
Option (b) is not possible, since if L1 is recursive, then
will also be recursive.
Theorem: If L1 and L2 is a pair of complementary language then either
• Both L1 and L2 are recursive.
• Neither L1, and L2 are recursively enumerable.
• One is recursively enumerable but not recursive, the other is not RE.
Option (b) is not possible, since if L1 is recursive, then
will also be recursive.Most Upvoted Answer
If L1 and L2 are a pair of complementary languages. Which of the follo...
Introduction:
In the theory of formal languages, a pair of languages L1 and L2 is said to be complementary if every string that belongs to one language does not belong to the other language. In other words, L1 and L2 do not have any common strings.
Explanation:
Let's analyze each option and determine if it is possible or not:
a) Both L1 and L2 are recursive:
If both languages are recursive, it means that there exists a Turing machine that can decide whether a given string belongs to L1 or L2. In this case, the languages are not complementary since the Turing machine can determine the membership of each string.
b) L1 is recursive and L2 is recursively enumerable but not recursive:
If L1 is recursive, it means that there exists a Turing machine that can decide whether a given string belongs to L1. On the other hand, if L2 is recursively enumerable but not recursive, it means that there exists a Turing machine that can enumerate all the strings in L2, but it cannot decide whether a given string belongs to L2.
In this case, the languages can be complementary because the Turing machine for L1 can decide whether a string belongs to L1 or not, and the Turing machine for L2 can enumerate all the strings in L2 without deciding their membership.
c) Neither L1 nor L2 is recursively enumerable:
If neither language is recursively enumerable, it means that there does not exist a Turing machine that can enumerate all the strings in L1 or L2. In this case, the languages can be complementary because there is no way to determine the membership of any string in either language.
d) One is recursively enumerable but not recursive, the other is not recursively enumerable:
If one language is recursively enumerable but not recursive, it means that there exists a Turing machine that can enumerate all the strings in that language, but it cannot decide whether a given string belongs to the language. If the other language is not recursively enumerable, it means that there does not exist a Turing machine that can enumerate all the strings in that language.
In this case, the languages can be complementary because the Turing machine for the recursively enumerable language can enumerate its strings without deciding their membership, and there is no Turing machine for the other language.
Conclusion:
From the above analysis, it can be concluded that option B is not possible. If L1 is recursive and L2 is recursively enumerable but not recursive, the languages can still be complementary.
In the theory of formal languages, a pair of languages L1 and L2 is said to be complementary if every string that belongs to one language does not belong to the other language. In other words, L1 and L2 do not have any common strings.
Explanation:
Let's analyze each option and determine if it is possible or not:
a) Both L1 and L2 are recursive:
If both languages are recursive, it means that there exists a Turing machine that can decide whether a given string belongs to L1 or L2. In this case, the languages are not complementary since the Turing machine can determine the membership of each string.
b) L1 is recursive and L2 is recursively enumerable but not recursive:
If L1 is recursive, it means that there exists a Turing machine that can decide whether a given string belongs to L1. On the other hand, if L2 is recursively enumerable but not recursive, it means that there exists a Turing machine that can enumerate all the strings in L2, but it cannot decide whether a given string belongs to L2.
In this case, the languages can be complementary because the Turing machine for L1 can decide whether a string belongs to L1 or not, and the Turing machine for L2 can enumerate all the strings in L2 without deciding their membership.
c) Neither L1 nor L2 is recursively enumerable:
If neither language is recursively enumerable, it means that there does not exist a Turing machine that can enumerate all the strings in L1 or L2. In this case, the languages can be complementary because there is no way to determine the membership of any string in either language.
d) One is recursively enumerable but not recursive, the other is not recursively enumerable:
If one language is recursively enumerable but not recursive, it means that there exists a Turing machine that can enumerate all the strings in that language, but it cannot decide whether a given string belongs to the language. If the other language is not recursively enumerable, it means that there does not exist a Turing machine that can enumerate all the strings in that language.
In this case, the languages can be complementary because the Turing machine for the recursively enumerable language can enumerate its strings without deciding their membership, and there is no Turing machine for the other language.
Conclusion:
From the above analysis, it can be concluded that option B is not possible. If L1 is recursive and L2 is recursively enumerable but not recursive, the languages can still be complementary.
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If L1 and L2 are a pair of complementary languages. Which of the following statement is not possible?a)Both L1, and L2 are recursive.b)L1 is recursive and L2 is recursively enumerable but not a recursive.c)Neither L1, nor L2 is recursively enumerable.d)One is recursively enumerable but not recursive, the other is is not recursively enumberable.Correct answer is option 'B'. Can you explain this answer?
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If L1 and L2 are a pair of complementary languages. Which of the following statement is not possible?a)Both L1, and L2 are recursive.b)L1 is recursive and L2 is recursively enumerable but not a recursive.c)Neither L1, nor L2 is recursively enumerable.d)One is recursively enumerable but not recursive, the other is is not recursively enumberable.Correct answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about If L1 and L2 are a pair of complementary languages. Which of the following statement is not possible?a)Both L1, and L2 are recursive.b)L1 is recursive and L2 is recursively enumerable but not a recursive.c)Neither L1, nor L2 is recursively enumerable.d)One is recursively enumerable but not recursive, the other is is not recursively enumberable.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If L1 and L2 are a pair of complementary languages. Which of the following statement is not possible?a)Both L1, and L2 are recursive.b)L1 is recursive and L2 is recursively enumerable but not a recursive.c)Neither L1, nor L2 is recursively enumerable.d)One is recursively enumerable but not recursive, the other is is not recursively enumberable.Correct answer is option 'B'. Can you explain this answer?.
If L1 and L2 are a pair of complementary languages. Which of the following statement is not possible?a)Both L1, and L2 are recursive.b)L1 is recursive and L2 is recursively enumerable but not a recursive.c)Neither L1, nor L2 is recursively enumerable.d)One is recursively enumerable but not recursive, the other is is not recursively enumberable.Correct answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about If L1 and L2 are a pair of complementary languages. Which of the following statement is not possible?a)Both L1, and L2 are recursive.b)L1 is recursive and L2 is recursively enumerable but not a recursive.c)Neither L1, nor L2 is recursively enumerable.d)One is recursively enumerable but not recursive, the other is is not recursively enumberable.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If L1 and L2 are a pair of complementary languages. Which of the following statement is not possible?a)Both L1, and L2 are recursive.b)L1 is recursive and L2 is recursively enumerable but not a recursive.c)Neither L1, nor L2 is recursively enumerable.d)One is recursively enumerable but not recursive, the other is is not recursively enumberable.Correct answer is option 'B'. Can you explain this answer?.
Solutions for If L1 and L2 are a pair of complementary languages. Which of the following statement is not possible?a)Both L1, and L2 are recursive.b)L1 is recursive and L2 is recursively enumerable but not a recursive.c)Neither L1, nor L2 is recursively enumerable.d)One is recursively enumerable but not recursive, the other is is not recursively enumberable.Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
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