GATE CS 2020 Set 1 — Question 18
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Theory of Computation → Finite Automata & Regular Languages → Closure Properties (Regular)
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Question
Consider the following statements.I. If is regular, then both and must be regular.
II. The class of regular languages is closed under infinite union.Which of the above statements is/are TRUE?
II. The class of regular languages is closed under infinite union.Which of the above statements is/are TRUE?
Correct answer
(D) Neither I nor II
Solution
Statement I is FALSE.
Let be a non-regular language (e.g., ) and be its complement . The union , which is regular. However, neither nor is regular.Statement II is FALSE.
The class of regular languages is closed under finite union, but not infinite union. For example, every singleton set is regular. The infinite union of singleton sets can form any language, including non-regular ones (e.g., , which is not regular).Thus, neither statement is true.
Let be a non-regular language (e.g., ) and be its complement . The union , which is regular. However, neither nor is regular.Statement II is FALSE.
The class of regular languages is closed under finite union, but not infinite union. For example, every singleton set is regular. The infinite union of singleton sets can form any language, including non-regular ones (e.g., , which is not regular).Thus, neither statement is true.
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