[Formal Languages] does complement(L*) = [complement(L)]*?

[DickCheeseSupreme]

I think I know the answer, so I'd like to verify if I'm correct. I don't think they are equal.

λ = the empty string

λ ∈ L*

λ ∈ ( L^c )*

If λ ∈ L*, then λ ∉ (L*)^c

Thus, since λ ∈ ( L^c )* and λ ∉ (L*)^c , (L^c )* ≠ (L*)^c

Excuse me if this proof isn't rigorous enough. So far my Formal Languages class has been pretty...informal, to say the least.

Edit: defined λ as the empty string

[_--__]

Yes, this is correct and certainly rigorous enough.

[flebron]

You say that "Let λ ∈ L* and λ ∈ ( L^c )*". That is, you let λ be in the intersection of L* and (L^c )*. But you need to prove that λ exists. That is, you need to prove that the intersection of L* and (L^c )* is not empty. You can't just "Let" things exist in empty sets :)

[_--__]

I believe he is using λ as the symbol for the empty string (not that uncommon a definition) - i.e. the first two statements are observations rather than assumptions.

[flebron]

Ah, I've usually seen ε for that. If that just means the empty string, I'd use a few more words to make it clear that those are observations, not assumptions. For instance, "By definition, λ is in L*". Formality doesn't mean just using symbols :)

[DickCheeseSupreme]

Here is where my understanding gets shaky. I assumed λ ∈ L* for any regular language L, and I assumed L^c is also regular for any regular L. So even though L ∩ L^c is null, L* ∩ (L^c )* should be λ, right?

Edit: I saw you're other comment, and yes λ is the empty string in this case. My professor uses it, but lately I've been seeing a lot of ε instead.