GATE CSE 1998 | Question: 2.6

Question

Which of the following statements is false?

1. Every finite subset of a non-regular set is regular
2. Every subset of a regular set is regular
3. Every finite subset of a regular set is regular
4. The intersection of two regular sets is regular

Comments

Is non-regular set closed under intersection.

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Non-regular may consist of 2 CFL and CFL’s are not closed under intersection. So the outcome may or may not be CFL. which eventually would give non-regular. So, generically they are closed. Please correct if I am wrong.

Answer 1

Correct Option: B

Any language is a subset of $\Sigma^*$ which is a regular set. So, if we take any non-regular language, it is a subset of a regular language.

(a) and (c) are regular as any finite language is regular.

(d) is regular as regular set is closed under intersection.

Comments

example of option B)a^nb^n is non regular subset of regular (a/b)*.

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Also, regular is not closed under subset, so the output may or may not be regular.

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$a^*b^*$ is regular but its subset $a^nb^n$ is not regular.

Answer 2

B is false because if u take a*b* as an example a^n b^n is a subset of a*b* but its not regular

Answer 3

(c) Every finite subset of a regular set is regular this is false

example:a^n b^n from regular set (a+b)* is not regular

Comments

No. The set you gave is infinite. Any finite set is trivially regular as there are only finite number of strings. (Finite class of languages is a subset of regular class).

Your example is for (b) and that is the false statement.

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sorry for it.it  is a silly mistake i thought it is a very easy question so i just took any option infront of me .it happens with me at times

(b) Every subset of a regular set is regular this is false