Question

Consider the following two statements about regular languages:
\(S_1:\) Every infinite regular language contains an undecidable language as a subset.
\(S_2:\) Every finite language is regular.
Which one of the following choices is correct?

• A. Only \(S_1\) is true.
• B. Only \(S_2\) is true.
• C. Both \(S_1\) and \(S_2\) are true.
• D. Neither \(S_1\) nor \(S_2\) is true.

Hint:

Regularity depends on recognizability by a finite automaton; finiteness always guarantees regularity.

Answer 1

The Correct Option is C

To determine the accuracy of the given statements \(S_1\) and \(S_2\), let's explore them one by one using concepts from the Theory of Computation.

1. **Statement \(S_1\):** Every infinite regular language contains an undecidable language as a subset.

   This statement is true. Here's why:

   Regular languages are closed under intersection with decidable sets and infinite languages will always contain infinite subsets. One characteristic of an infinite regular language is that it can represent all possible strings over an alphabet as each string can be accepted using a finite automaton. Since undecidable languages can be subsets of these infinite sets without losing regularity, and given the vast number of subsets in an infinite set, at least one undecidable language can indeed be a subset of every infinite regular language.

2. **Statement \(S_2\):** Every finite language is regular.

   This statement is also true. Here's why:

   A language is regular if it can be represented by a finite automaton or described using a regular expression. A finite language is one with a finite number of strings, all of which can be represented by a finite automaton that explicitly lists each string as part of its accepting states. As such, every finite language can indeed be represented by a regular expression or a DFA (Deterministic Finite Automaton), hence it is regular.

Conclusion: Both statements \(S_1\) and \(S_2\) are true, making the correct choice: Both \(S_1\) and \(S_2\) are true.