I had a test a couple of days ago and one of the question had 2 statements :$L^+ = L^\ast$$L$ contains $\varepsilon$I had to say does 1 imply 2, does 2 imply 1 or do they both imply each other.I answered that only 2 implies 1 but the answer is that they both imply each other.But I don't understand, if $L = \emptyset$ (Empty Language), then $L^\ast = \{\varepsilon\}$ and $L^+ = \{ \varepsilon \}$, they are equal and $L$ doesn't contain $\varepsilon$.Am I wrong?
Kleene star Empty language
formal languages;kleene star
You're wrong about $L^+$: $L = \emptyset \rightarrow L^+ = \emptyset$, but $L = \emptyset \rightarrow L^* = \{\epsilon\}$