Regular and not regular Language

[mathmari]

Hey!

If $K$ is a set of natural numbers and $b$ is a natural number greater than $1$, let $$L_b(K)=\{w \mid w \text{ is the representation in base } b \text{ of some number in } K\}$$
Leading $0$s are not allowed in the representation of a number.
For example, $L_2(\{3, 5\})=\{11, 101\}$ and $L_3(\{3, 5\})=\{10, 12\}$.
Give an example of a set $K$ for which $L_2(K)$ is regular but $L_3(K)$ is not regular. Prove that your example works.

I am confused... How can $L_i$ be regular when $L_j$ is not regular? (Wondering)

Could you explain it to me? (Wondering)

 

[Evgeny.Makarov]

Congratulations on your 1000 posts!

There is a page on CS StackExchange about your question, but it requires some effort to grasp.

 

[mathmari]

Congratulations on your 1000 posts!

Thank you! (Happy)
For the set $K=\{2^i, i\in \mathbb{N}\}=\{2,4,8, \dots \}$ we have $L_2(K)=\{10,100,1000, \dots \}$ which is regular since it is accepted by the regular expression $10^+$, right?

So we have to show that for this set $L_3(K)$ is not regular. To do that do we have to apply the pumping lemma? Or is there also an other way?

 

[Evgeny.Makarov]

For the set $K=\{2^i, i\in \mathbb{N}\}=\{2,4,8, \dots \}$ we have $L_2(K)=\{10,100,1000, \dots \}$ which is regular since it is accepted by the regular expression $10^+$, right?

Yes.

So we have to show that for this set $L_3(K)$ is not regular. To do that do we have to apply the pumping lemma? Or is there also an other way?

The StackExchange page contains a couple of proofs, but I have not studied them carefully. In my opinion, this problem goes somewhat beyond the necessary minimum a student is supposed to take from the theory of computation course. It is more like a challenge problem. Since what I am prepared to give is a "marginal legal advice" (a slogan from a popular American radio show about legal matters), I can't promise further help with this problem.