Kleene star operations

Question

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

Answer 1

Hint:

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 $L_1\Sigma^{*}L_2=\Sigma^{*}L_2=\Sigma^{*}$.

Answer 2

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.