GATE CS 2021 Set 2 — Question 46
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Theory of Computation → Finite Automata & Regular Languages → Chomsky Hierarchy
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Question
Consider the following two statements about regular languages:
: Every infinite regular language contains an undecidable language as a subset.
: Every finite language is regular.
Which one of the following choices is correct?
: Every infinite regular language contains an undecidable language as a subset.
: Every finite language is regular.
Which one of the following choices is correct?
Correct answer
(C) Both S₁ and S₂ are true.
Solution
is true: Any infinite set (including an infinite regular language) has uncountably many subsets (). Since the set of all decidable languages is countable, there must be uncountably many subsets that are undecidable.
is true: Every finite language is regular because a finite automaton can be constructed to accept exactly the finite set of strings in the language (or a regular expression can be formed by the union of all strings).
is true: Every finite language is regular because a finite automaton can be constructed to accept exactly the finite set of strings in the language (or a regular expression can be formed by the union of all strings).
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