I have to prove or disprove the implications in these two situations
- $L^* = \emptyset$ $\rightarrow$ $L$ is infinite
- $L^+ = \emptyset$ $\rightarrow$ $L$ is infinite
Here are my thoughts.
I would say Kleene star operation (*) is union of
$\qquad L^0 \cup L^1 \cup L^2 \cup \dots$
and since $\_^0 = \{\epsilon\}$ (even $\emptyset^ 0 = \{\epsilon\}$), there is no language $L$ such that $L^* = \emptyset$.
How to prove finiteness/infiniteness?
Only language $L$ where $L^+ = \emptyset$ is $\emptyset$ which is a finite language. But how to prove there is no other language that is inifinite and the argument is valid for it?
EDIT: will it help if I rewrite the implications $A \rightarrow B$ to $\neg B \rightarrow \neg A$ ?
- $L$ is finite $\rightarrow$ $L^* \neq \emptyset$
- $L$ is finite $\rightarrow$ $L^+ \neq \emptyset$
2) can be now interpreted as $\emptyset^+ = \{\epsilon\}$ and $\emptyset$ is a finite language. So the implication doesn't work.