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Consider the following statements :
S1 : DCFL's are closed under complement
S2 : DCFL's are closed under Homomorphism
S3 : DCFL's are closed under union with regular
Which option is correct ?
- a)Only S1 is true
- b)Only S1 and S2 are true
- c)Only S2 and S3 are true
- d)Only S1 and S3 are true
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Consider the following statements :S1 : DCFL's are closed under co...
S1 : True
S2 : DCFL are closed under inverse homomorphism. Hence False
S3 : Union , intersection and subtraction with regular is always closed. Hence True
So only S1 and S3 are true.
Most Upvoted Answer
Consider the following statements :S1 : DCFL's are closed under co...
Statement Analysis:
Let's analyze each statement one by one.
S1: DCFLs are closed under complement.
A language is said to be closed under complement if the complement of the language is also in the same class of languages. In other words, if a language L is in a certain class of languages, then the complement of L is also in the same class.
For this statement to be true, we need to show that if a language is a deterministic context-free language (DCFL), then its complement is also a DCFL.
S2: DCFLs are closed under homomorphism.
A language is said to be closed under homomorphism if the homomorphic image of the language is also in the same class of languages. In other words, if a language L is in a certain class of languages, then the homomorphic image of L is also in the same class.
For this statement to be true, we need to show that if a language is a DCFL, then its homomorphic image is also a DCFL.
S3: DCFLs are closed under union with regular.
A language is said to be closed under union with regular languages if the union of the language with any regular language is also in the same class of languages. In other words, if a language L is in a certain class of languages, then the union of L with any regular language is also in the same class.
For this statement to be true, we need to show that if a language is a DCFL, then its union with any regular language is also a DCFL.
Analysis:
- S1: DCFLs are closed under complement. This statement is true. The complement of a DCFL is also a DCFL. Therefore, S1 is true.
- S2: DCFLs are closed under homomorphism. This statement is false. DCFLs are not closed under homomorphism. The homomorphic image of a DCFL may not be a DCFL.
- S3: DCFLs are closed under union with regular. This statement is true. The union of a DCFL with any regular language is also a DCFL. Therefore, S3 is true.
Based on the analysis, the correct option is Only S1 and S3 are true (Option D).
Let's analyze each statement one by one.
S1: DCFLs are closed under complement.
A language is said to be closed under complement if the complement of the language is also in the same class of languages. In other words, if a language L is in a certain class of languages, then the complement of L is also in the same class.
For this statement to be true, we need to show that if a language is a deterministic context-free language (DCFL), then its complement is also a DCFL.
S2: DCFLs are closed under homomorphism.
A language is said to be closed under homomorphism if the homomorphic image of the language is also in the same class of languages. In other words, if a language L is in a certain class of languages, then the homomorphic image of L is also in the same class.
For this statement to be true, we need to show that if a language is a DCFL, then its homomorphic image is also a DCFL.
S3: DCFLs are closed under union with regular.
A language is said to be closed under union with regular languages if the union of the language with any regular language is also in the same class of languages. In other words, if a language L is in a certain class of languages, then the union of L with any regular language is also in the same class.
For this statement to be true, we need to show that if a language is a DCFL, then its union with any regular language is also a DCFL.
Analysis:
- S1: DCFLs are closed under complement. This statement is true. The complement of a DCFL is also a DCFL. Therefore, S1 is true.
- S2: DCFLs are closed under homomorphism. This statement is false. DCFLs are not closed under homomorphism. The homomorphic image of a DCFL may not be a DCFL.
- S3: DCFLs are closed under union with regular. This statement is true. The union of a DCFL with any regular language is also a DCFL. Therefore, S3 is true.
Based on the analysis, the correct option is Only S1 and S3 are true (Option D).
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Consider the following statements :S1 : DCFL's are closed under complementS2 : DCFL's are closed under HomomorphismS3 : DCFL's are closed under union with regularWhich option is correct ?a)Only S1 is trueb)Only S1 and S2 are truec)Only S2 and S3 are trued)Only S1 and S3 are trueCorrect answer is option 'D'. Can you explain this answer?
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Consider the following statements :S1 : DCFL's are closed under complementS2 : DCFL's are closed under HomomorphismS3 : DCFL's are closed under union with regularWhich option is correct ?a)Only S1 is trueb)Only S1 and S2 are truec)Only S2 and S3 are trued)Only S1 and S3 are trueCorrect answer is option 'D'. 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 Consider the following statements :S1 : DCFL's are closed under complementS2 : DCFL's are closed under HomomorphismS3 : DCFL's are closed under union with regularWhich option is correct ?a)Only S1 is trueb)Only S1 and S2 are truec)Only S2 and S3 are trued)Only S1 and S3 are trueCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statements :S1 : DCFL's are closed under complementS2 : DCFL's are closed under HomomorphismS3 : DCFL's are closed under union with regularWhich option is correct ?a)Only S1 is trueb)Only S1 and S2 are truec)Only S2 and S3 are trued)Only S1 and S3 are trueCorrect answer is option 'D'. Can you explain this answer?.
Consider the following statements :S1 : DCFL's are closed under complementS2 : DCFL's are closed under HomomorphismS3 : DCFL's are closed under union with regularWhich option is correct ?a)Only S1 is trueb)Only S1 and S2 are truec)Only S2 and S3 are trued)Only S1 and S3 are trueCorrect answer is option 'D'. 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 Consider the following statements :S1 : DCFL's are closed under complementS2 : DCFL's are closed under HomomorphismS3 : DCFL's are closed under union with regularWhich option is correct ?a)Only S1 is trueb)Only S1 and S2 are truec)Only S2 and S3 are trued)Only S1 and S3 are trueCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statements :S1 : DCFL's are closed under complementS2 : DCFL's are closed under HomomorphismS3 : DCFL's are closed under union with regularWhich option is correct ?a)Only S1 is trueb)Only S1 and S2 are truec)Only S2 and S3 are trued)Only S1 and S3 are trueCorrect answer is option 'D'. Can you explain this answer?.
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