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We define the minimal words language of $L, \min(L)$, to be the language of all words in $L$ that don't have any prefix in $L$.

Assume $L$ is regular language. I need to prove by building an automaton that $\min(L)$ is regular.

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    Look, for example, at the edits I just made. Click on the link after the word "edited" and then look at the source, by clicking on "source".2012-11-30

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Hint: Consider a deterministic automaton, if you happen to be at the accepting state, anything that comes later should send you to the "trash" state.