Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Which of the following statements is false?a)...
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Which of the following statements is false?
- a)Every finite subset of a non-regular set is regular
- b)Every subset of a regular set is regular
- c)Every finite subset of a regular set is regular
- d)The intersection of two regular sets is regular
Correct answer is option 'B'. Can you explain this answer?
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Which of the following statements is false?a)Every finite subset of a ...
Any language is a subset of ∑* which is a regular set. So, if we take any non-regular language, it is a subset of a regular language.
(a) and (c) are regular as any finite language is regular.
(d) is regular as regular set is closed under intersection.
Most Upvoted Answer
Which of the following statements is false?a)Every finite subset of a ...
False Statement: Every subset of a regular set is regular.
Explanation:
To understand why this statement is false, we need to have a clear understanding of what a regular set is.
Regular Set:
In formal language theory, a regular set is a set of strings that can be recognized by a finite-state automaton. In simple terms, a regular set is a set of strings that can be generated by a regular expression or accepted by a deterministic finite automaton (DFA).
Counterexample:
To disprove the statement, we need to find a counterexample where a subset of a regular set is not regular. Let's consider the following example:
Regular Set: L = {a^n b^n | n ≥ 0}
Subset: S = {ε, ab}
Explanation of the Example:
In the regular set L, every string consists of 'a's followed by the same number of 'b's. For example, "ab", "aabb", "aaabbb", etc., are all part of the regular set L.
Now, let's consider the subset S, which contains two strings: ε (empty string) and "ab". Both these strings are part of the regular set L.
However, the subset S is not regular because it violates the rule of having an equal number of 'a's and 'b's. The string "ab" does not have an equal number of 'a's and 'b's, which means it cannot be recognized by a finite-state automaton, and hence, it is not a regular set.
Conclusion:
The example above demonstrates that there exists a subset of a regular set (L) that is not regular (S). Therefore, the statement "Every subset of a regular set is regular" is false.
Explanation:
To understand why this statement is false, we need to have a clear understanding of what a regular set is.
Regular Set:
In formal language theory, a regular set is a set of strings that can be recognized by a finite-state automaton. In simple terms, a regular set is a set of strings that can be generated by a regular expression or accepted by a deterministic finite automaton (DFA).
Counterexample:
To disprove the statement, we need to find a counterexample where a subset of a regular set is not regular. Let's consider the following example:
Regular Set: L = {a^n b^n | n ≥ 0}
Subset: S = {ε, ab}
Explanation of the Example:
In the regular set L, every string consists of 'a's followed by the same number of 'b's. For example, "ab", "aabb", "aaabbb", etc., are all part of the regular set L.
Now, let's consider the subset S, which contains two strings: ε (empty string) and "ab". Both these strings are part of the regular set L.
However, the subset S is not regular because it violates the rule of having an equal number of 'a's and 'b's. The string "ab" does not have an equal number of 'a's and 'b's, which means it cannot be recognized by a finite-state automaton, and hence, it is not a regular set.
Conclusion:
The example above demonstrates that there exists a subset of a regular set (L) that is not regular (S). Therefore, the statement "Every subset of a regular set is regular" is false.
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Which of the following statements is false?a)Every finite subset of a non-regular set is regularb)Every subset of a regular set is regularc)Every finite subset of a regular set is regulard)The intersection of two regular sets is regularCorrect answer is option 'B'. Can you explain this answer?
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