Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Which of the following statements is incorrec...
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Which of the following statements is incorrect?
- a)Union of two DCFL is a CFL.
- b)Complementation of a regular language is a regular language.
- c)Turing recognizable languages are closed under intersection.
- d)None.
Correct answer is option 'D'. Can you explain this answer?
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Which of the following statements is incorrect?a)Union of two DCFL is ...
Statement 1 is correct because union of two DCFL may or may not be DCFL but surely CFL. Statement 2 is correct because regular languages are closed under complementation. Statement 3 is correct because Turing recognizable languages or recursive enumerable languages are closed under intersection.
Most Upvoted Answer
Which of the following statements is incorrect?a)Union of two DCFL is ...
Incorrect Statement: None (Option D)
Explanation:
- The incorrect statement is option D, which states that none of the statements are incorrect. However, this is not true as there are incorrect statements among the given options.
Analysis of each statement:
a) Union of two DCFL is a CFL:
- This statement is correct. The union of two deterministic context-free languages (DCFL) is a context-free language (CFL). This can be proven by constructing a context-free grammar or a pushdown automaton that recognizes the union of the two languages.
b) Complementation of a regular language is a regular language:
- This statement is correct. The complement of a regular language is also a regular language. This can be proven by constructing a deterministic finite automaton (DFA) for the original language and then flipping the accept and non-accept states to obtain a DFA for the complement language.
c) Turing recognizable languages are closed under intersection:
- This statement is incorrect. Turing recognizable (also known as recursively enumerable) languages are not closed under intersection. This means that the intersection of two Turing recognizable languages may not be Turing recognizable. This can be proven by showing a reduction from the halting problem to the intersection of two Turing recognizable languages.
Conclusion:
- The correct answer is option D, as it incorrectly states that none of the statements are incorrect. However, statement c) is incorrect as Turing recognizable languages are not closed under intersection.
Explanation:
- The incorrect statement is option D, which states that none of the statements are incorrect. However, this is not true as there are incorrect statements among the given options.
Analysis of each statement:
a) Union of two DCFL is a CFL:
- This statement is correct. The union of two deterministic context-free languages (DCFL) is a context-free language (CFL). This can be proven by constructing a context-free grammar or a pushdown automaton that recognizes the union of the two languages.
b) Complementation of a regular language is a regular language:
- This statement is correct. The complement of a regular language is also a regular language. This can be proven by constructing a deterministic finite automaton (DFA) for the original language and then flipping the accept and non-accept states to obtain a DFA for the complement language.
c) Turing recognizable languages are closed under intersection:
- This statement is incorrect. Turing recognizable (also known as recursively enumerable) languages are not closed under intersection. This means that the intersection of two Turing recognizable languages may not be Turing recognizable. This can be proven by showing a reduction from the halting problem to the intersection of two Turing recognizable languages.
Conclusion:
- The correct answer is option D, as it incorrectly states that none of the statements are incorrect. However, statement c) is incorrect as Turing recognizable languages are not closed under intersection.
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Which of the following statements is incorrect?a)Union of two DCFL is a CFL.b)Complementation of a regular language is a regular language.c)Turing recognizable languages are closed under intersection.d)None.Correct answer is option 'D'. Can you explain this answer?
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