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I had a test a couple of days ago and one of the question had 2 statements :

  1. $L^+ = L^\ast$

  2. $L$ 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?

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You're wrong about $L^+$:

$L = \emptyset \rightarrow L^+ = \emptyset$, but $L = \emptyset \rightarrow L^* = \{\epsilon\}$

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