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GATE CSE 2021 Set 2 Q46: Undecidable subsets of infinite regular languages; finite languages

Question

Consider the following two statements about regular languages:

S1S_1: Every infinite regular language contains an undecidable language as a subset.

S2S_2: Every finite language is regular.

Which one of the following choices is correct?

Try it: multiple choice: exactly one correct option

Select one option
Wrong answer costs 2/3 mark in GATE.
Show answer and explanation

Answer: C (official GATE 2021 final answer key)

Concept used: An infinite set has uncountably many subsets but there are only countably many Turing machines.

First step: Count: an infinite language has 2ℵ02^{\aleph_0} subsets, while decidable languages are countable; and a finite language is a finite union of single strings.

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Source: official GATE 2021 Set 2 CS question paper and final answer key, organised by IIT Bombay. Question paper · Answer key

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