GATE CS 2024 Set 1 — Question 23
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Theory of Computation → Finite Automata & Regular Languages → Closure Properties (Regular)
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Question
Let be two regular languages and a language which is not regular. Which of the following statements is/are always TRUE?
Correct answer
(C) L₃ is not regular; (D) L₁ ∪ L₂ is regular
Solution
Let be regular languages and be a non-regular language.
- Option (A): if and only if . This is false. is equivalent to . For to be true, we also need (i.e., ).
- Option (B): is not regular. This is false. For example, if (which is regular) and is any non-regular language, then , which is regular.
- Option (C): is not regular. This is true. Regular languages are closed under complementation. If were regular, then its complement would also be regular, which contradicts the given that is not regular.
- Option (D): is regular. This is true. Regular languages are closed under complementation and union. Since and are regular, and are regular, and their union is also regular.
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