GATE CSE 1990 | Question: 3-viii

Question

Let $R_{1}$ and $R_{2}$ be regular sets defined over the alphabet $\Sigma$ Then:

• A. $R_{1} \cap R_{2}$ is not regular.
• B. $R_{1} \cup R_{2}$ is regular.
• C. $\Sigma^{*}-R_{1}$ is regular.
• D. $R_{1}^{*}$ is not regular.

Comments

Regular language is closed under

• Union
• Intersection
• Kleene Closure
• Concatenation
• Complementation
• Difference
• Reversal

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∑^∗ 

^{is  regular and if remove something then also it will remain regular...}

^{Regular languages are closed everywhere}

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if L1 is regular and L2 is regular then L1 U L2 is regular (one way)

if they ask L1 U L2 is regular then what are your comments about L1 and L2 , then we can;t say comment on them

also, in case of union and intersection with regular, upgradatoin is not possible

e.g) R ^ CFL = CSL and not CFL as CFL are not closed under intersection

also, CFL and RE are not closed under complement

Answer 1

Regular Languages are closed under

1. Intersection
2. Union
3. Complement
4. Kleen-Closure

$\Sigma^∗−R_1$ is the complement of $R_1$
 

Correct Options: B;C

Answer 2

1. R1 ∩ R2 is not regular.-FALSE,Regular sets are closed under Intersection
2. R1 ∪ R2 is regular. TRUE,Regular sets are closed under Union
3. ∑^∗−R1 is regular,TRUE,Regular sets are closed under Complement
4. R_1^* is not regular.FALSE,Regular sets are closed under Intersection

Comments

Everything is true if they have not used the word not in the question.

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4)R1* is not Regular (FALSE) because regular language is closed under kleen closure.

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then who told, "Choose the correct alternatives (More than one may be correct)" ?