Recognize the language

Question

If L be a language recognizable by a finite automata, then language from {L}={w such that w is prefix of v where v belongs to L},is a

a. Regular Language

b. Context Free language

c. Context Sensitive Language

d. Recursive Enumerable Language

Comments

a) regular language

regular languages are closed under prefix operation..

Answer 1

Prefix of a langauge are finite. Therefore, {L} will be regular language. Option (A)

Comments

Wrong. Prefix language of any infinite language will be infinite.

Answer 2

Take the DFA and convert all the states as accepting ones for accepting all prefixes.

If you want only the proper prefixes convert all the states to accepting ones, then exclude  the initial state and the final states, but then include the states that are one transition away from any other final state.

Comments

 

Proper prefixes are the prefixes excluding the string and ϵ. 

But for the language?

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Well, prefixes for the strings in the language, right? The language formed by the prefixes of the strings in the original language.

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Thats for “prefix”, not “proper prefix”. Anyway you can check that definition and the answer you have given once more.