Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Which of the following is true?a)All subsets ...
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Which of the following is true?
- a)All subsets of a regular set are always regular
- b)All finite subsets of non-regular set are always regular
- c)Union of two non regular set of language is not regular
- d)Infinite times union of finite set is always regular
Correct answer is option 'B'. Can you explain this answer?
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Which of the following is true?a)All subsets of a regular set are alwa...
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Which of the following is true?a)All subsets of a regular set are alwa...
Explanation:
Regular set is defined as a set of strings that can be generated by a regular grammar or recognized by a finite automaton. Now, let's analyze each option to find the true statement.
Option A: All subsets of a regular set are always regular.
This statement is not true. A counterexample can be a regular set A = {a, ab}, whose subsets are A1 = {ε}, A2 = {a}, A3 = {ab}, A4 = {a, ab}. Here, A3 and A4 are not regular sets.
Option B: All finite subsets of non-regular set are always regular.
This statement is true. A non-regular language is one that cannot be generated by a regular grammar or recognized by a finite automaton. Therefore, any finite subset of a non-regular set can only contain finite strings that can be generated or recognized by a finite automaton, and hence, it is always regular.
Option C: Union of two non-regular sets of language is not regular.
This statement is not true. A counterexample can be two non-regular sets A = {a^n b^n | n ≥ 0} and B = {a^n b^n c^n | n ≥ 0}. Both these sets are not regular, but their union A ∪ B = {a^n b^n | n ≥ 0} ∪ {a^n b^n c^n | n ≥ 0} = {a^n b^n c^m | n, m ≥ 0} is a regular set.
Option D: Infinite times union of finite sets is always regular.
This statement is not true. A counterexample can be an infinite set A = {a^n | n ≥ 0}, whose finite subsets are all regular sets. But, the infinite union of all these finite subsets gives the set A itself, which is not regular. Therefore, the infinite times union of finite sets is not always regular.
Hence, the correct option is B.
Regular set is defined as a set of strings that can be generated by a regular grammar or recognized by a finite automaton. Now, let's analyze each option to find the true statement.
Option A: All subsets of a regular set are always regular.
This statement is not true. A counterexample can be a regular set A = {a, ab}, whose subsets are A1 = {ε}, A2 = {a}, A3 = {ab}, A4 = {a, ab}. Here, A3 and A4 are not regular sets.
Option B: All finite subsets of non-regular set are always regular.
This statement is true. A non-regular language is one that cannot be generated by a regular grammar or recognized by a finite automaton. Therefore, any finite subset of a non-regular set can only contain finite strings that can be generated or recognized by a finite automaton, and hence, it is always regular.
Option C: Union of two non-regular sets of language is not regular.
This statement is not true. A counterexample can be two non-regular sets A = {a^n b^n | n ≥ 0} and B = {a^n b^n c^n | n ≥ 0}. Both these sets are not regular, but their union A ∪ B = {a^n b^n | n ≥ 0} ∪ {a^n b^n c^n | n ≥ 0} = {a^n b^n c^m | n, m ≥ 0} is a regular set.
Option D: Infinite times union of finite sets is always regular.
This statement is not true. A counterexample can be an infinite set A = {a^n | n ≥ 0}, whose finite subsets are all regular sets. But, the infinite union of all these finite subsets gives the set A itself, which is not regular. Therefore, the infinite times union of finite sets is not always regular.
Hence, the correct option is B.
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Which of the following is true?a)All subsets of a regular set are always regularb)All finite subsets of non-regular set are always regularc)Union of two non regular set of language is not regulard)Infinite times union of finite set is always regularCorrect answer is option 'B'. Can you explain this answer?
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