To prove that regular languages are closed under f $($L$),$ where

f $($L$)=\{$w in L $|$ no proper prefix of w is in L$\},$

$1$

we can use existing closure properties, build a DFA and prove it does recognize

the language.

Let P be the language which consists of proper prefixes of all strings in L$,.$e$.,$

P $=\{$w in L $|$ all proper prefixes of w$\}$

P $=$

$($

w in L

prefixes$($w$)$

$)$

$.$ Here, prefixes$($w$)$ represents the set of all proper prefixes of $($w$).$

f $($L$)$ can be expressed as the set difference of L and P$.$

f $($L$)=$ L$\backslash$P

$1.$ Construct a DFA, M $$ for P :

$$ For each string w in L$,$ modify the DFA for L to accept all proper prefixes

of w$.$ This can be achieved by marking the states that correspond to

proper prefixes as accepting states.

$$ The resulting DFA recognizes the set of all strings with a proper prefix in

L$.$

$$ Construction of M $$:

$1.$ States: Q$=$

$2.$ Initial State:

$0=$

$3.$ Accepting States: $.$

$4.$ Transition Function:

since L and P are regular, f$($L$)$ is also regular since regular languages are closed under set difference