disprove that $(L \setminus \{\epsilon\})^* = L^* \setminus \{\epsilon\}$

Question

I need to disprove that for every formal language: $$(L \setminus \{\epsilon\})^* = L^* \setminus \{\epsilon\}$$ ($\epsilon$ is the empty word). I don't have an example to show that it isn't true, so can you provide me of an example that it is a false statement? thanks a lot.

Answer 1

The language $(L \setminus \{\epsilon\})^*$ always contains the empty word, but the language $L^* \setminus \{\epsilon\}$ never contains it. Thus they are always different.