True / False (Regular Languages)

Question

1. $L_{1} L_{2}$ is regular if and only if $L_{1}$ and $L_{2}$ both are regular
2. If $L_{1}\cup L_{2}$ is regular and $L_{2}$ is finite then $L_{1}$  is regular
3. If $L_{1}\cap L_{2}$ is regular then $L_{2}$ and $L_{1}$ may or may not be regular
4. If $L_{1}$$L_{2}$ is regular then $L_{2}$ and $L_{1}$ are regular

Comments

let $L_1=a^nb^n,L2=\phi$

1. and 4. False

$L_1 L_2=a^nb^n \,.\, \phi=\phi=\text{regular}$

Here $L_1 \text{is not regular}$

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2 and 3 are true

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ok and one more question please explain

If L1 ∪ L2 is regular and L1 is regular, then L2 is regular.

This is false but why?

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$L_1=(a+b)^{*} ,\text{regular}$

$L_2=a^{n}b^{n},\text{a CFL}$

$L_1 \cup L_2=(a+b)^{*},\text{a regular}$

here $L_2$ is nor regular hence false

Answer 1

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Comments

explanation of 3rd is quite beautiful.

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i think 4th explanation is wrong

it is concatenation not intersection

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¢ . a^p = ¢ ( fie)

it is also true

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ohh thanks

Answer 2

1 . False

2. True

3. True

4. False