GATE CSE 2020 Q18: Regularity of a union and closure under infinite union

Consider the following statements.

I. If $L_1 \cup L_2$ is regular, then both $L_1$ and $L_2$ must be regular.

II. The class of regular languages is closed under infinite union.

Which of the above statements is/are TRUE?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: D (official GATE 2020 final answer key)

Concept used: Closure properties hold for finite unions only; a regular union says nothing about its parts.

First step: Take $L_1 = \{a^n b^n\}$ and $L_2 = \Sigma^*$ for I; for II, write $\{a^n b^n\}$ as an infinite union of singleton languages.

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Source: official GATE 2020 CS question paper and final answer key, organised by IIT Delhi. Question paper · Answer key