Tomorrow I have my exam and I have still some doubts about some of the following TRUE/FALSE statements about REGULAR LANGUAGES.
Can someone help me and explain me why?
1) For all languages $L_{1}$ and $L_{2}$, if $L_{1} \subseteq L_{2}$, then $L_{1}^{*} \subseteq L_{2}^{*} $, where $L_{1}\neq L_{2}$.
2) For all languages $L_{1}$ and $L_{2}$, if $L_{1} \cap L_{2} = \emptyset $ and $L_{1} \cup L_{2} = \Sigma^{*}$ (the alphabet of $L_1\text{ and }L_2$, then $L_{1} = \overline{L_{2}}$, i.e., the complement of $L_2$.
3) If $L_{1}$ and $L_{2}$ are regular languages, then $(L_{1} \cap L_{2})^{*}\subseteq L_{1}^{*} \cap L_{2}^{*}$.
4) If L is a context-free language, then $L \setminus \{ \epsilon \}$ (where $\epsilon$ is empty string) is a context free grammar.
My answer: TRUE
Reason: if $L$ is a context-free language and $D$ is regular (in our case the Empty String which by definition is a regular language) then their difference is context-free languages.
5) If $L \setminus \{\epsilon\}$ is a regular language, then L is a regular language
My answer: TRUE
Observe that $L \setminus M = L \cap \overline{M}$. We already know that regular languages are closed under complement and intersection.