# Kleene star operations

Let $$𝚺$$ be any alphabet and let $$𝑳_𝟏 \subseteq 𝚺^{∗}$$ and $$𝑳_2 \subseteq 𝚺^{∗}$$ be two non-empty languages.

a. If $$𝑳_𝟏 𝚺^{∗} \neq 𝚺^{∗}$$ than what can we say about $$L_1$$.

b.Let $$\Lambda \in L_1$$ and $$\Lambda \in L_2$$. Show using axioms and theorems of languages that $$𝑳_𝟏 𝚺^{∗}𝑳_2 = 𝚺^{∗}$$

For (a), $$\Lambda$$ should not belong to $$L_1$$ but I do not know how to prove that.

For (b),we have to prove that $$𝑳_𝟏 𝚺^{∗}𝑳_2 \subseteq 𝚺^{∗}$$ and $$𝚺^{∗} \subseteq𝑳_𝟏 𝚺^{∗}𝑳_2$$ for equality to exist. We can also distinguish two cases, when $$L = \Lambda$$, then $$\Lambda 𝚺^{∗} \Lambda = 𝚺^{∗}$$, but how can we prove that when $$L \neq \Lambda$$

Any idea

• By the way, the culture here is one question per post. Expect a response saying that. – Rick Decker Feb 13 '20 at 1:09

You are right on (a). The proof involves key results for Kleene star as follows. $$\Sigma^{*}=\bigcup_{n\geqslant1}\Sigma^{n}\quad\text{and}\quad L\Sigma^{*}=\bigcup_{n\geqslant1}L\Sigma^{n}\quad\text{and}\quad \Sigma^{*}L=\bigcup_{n\geqslant1}\Sigma^{n}L$$ Where $$\Sigma^{n}$$ can be considered as the set of the concatenation of i strings in $$\Sigma$$, and $$L\subset \Sigma^{*}$$.
For (a), $$L_1\Sigma^{*}\subset\Sigma^{*}$$ since for all $$n,L_1\Sigma^{n}\subset\Sigma^{n}$$. If $$\Lambda\in L_1$$, then $$n,L_1\Sigma^{n}=\Sigma^{n}$$ for all $$n$$. Thus $$L_1\Sigma^{*}=\Sigma^{*}$$.
For (b), notice that if $$\Lambda\in L_1$$ and $$L_2$$, then $$L_1\Sigma^{*}=\Sigma^{*}$$ and $$\Sigma^{*}L_2=\Sigma^{*}$$. Thus $$L_1\Sigma^{*}L_2=\Sigma^{*}L_2=\Sigma^{*}$$.
You already have part of the answer for (a), but there's a bit more. For example, if $$L_1 = \{a\}$$ (and $$\Sigma$$ isn't just $$a$$) we certainly don't have $$L_1\Sigma^* = \Sigma^*$$. Can you come up with an extension? I knew you could.