GATE CS 2016 Set 2 — Question 52
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Theory of Computation → Finite Automata & Regular Languages → Closure Properties (Regular)
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Question
Consider the following two statements:I. If all states of an NFA are accepting states then the language accepted by the NFA is .
II. There exists a regular language such that for all languages , is regular.Which one of the following is CORRECT?
II. There exists a regular language such that for all languages , is regular.Which one of the following is CORRECT?
Correct answer
(B) Only II is true
Solution
Statement I is False:
An NFA accepts a string if there exists at least one path from the start state to an accepting state on input . Even if all states are accepting, it is possible that for some input string, there are no valid transitions defined (i.e., the path "dies"). In such a case, the string is not accepted. For example, an NFA with one state (accepting) and no transitions accepts only , not .Statement II is True:
We need to find one regular language such that is regular for any language . Consider the empty language . Since is regular, and for any language , , the result is always regular. Alternatively, any finite language works because the intersection of a finite language with any language is finite, and all finite languages are regular.Thus, only statement II is true.
An NFA accepts a string if there exists at least one path from the start state to an accepting state on input . Even if all states are accepting, it is possible that for some input string, there are no valid transitions defined (i.e., the path "dies"). In such a case, the string is not accepted. For example, an NFA with one state (accepting) and no transitions accepts only , not .Statement II is True:
We need to find one regular language such that is regular for any language . Consider the empty language . Since is regular, and for any language , , the result is always regular. Alternatively, any finite language works because the intersection of a finite language with any language is finite, and all finite languages are regular.Thus, only statement II is true.
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