Question

Consider the following statements:

I. If $L_1\cup L_2$ is regular, then both $L_{1}and{L}_2$ must be regular.

II. The class of regular languages is closed under infinite union.

Which of the above statements is/are TRUE?

Answer 1

If $L_1\cup L_2$ is regular, then neither needs to be regular.

Example $\left\{a^n{b}^n\right\}\cup {\left\{a^n{b}^n\right\}}^c=(a+b{)}^{\ast }$ is regular but $\left\{a^n{b}^n\right\}$ and its complement both are non-regular.

So statement I is false

The class of regular language is not closed under infinite union.

Proof: It is was closed under infinite union then

$a^n{b}^n=\left\{ϵ\right\}\cup \left\{ab\right\}\cup \left\{aabb\right\}\cup .........$ will be infinite union of finite languages (which are regular) and hence will become regular. But we know that $\{a^n{b}^n\mid n\ge 0\}$ is non-regular.

So II is false

So option (a). neither I nor II is the correct answer