Regular languages closed under prefix operation

Question

Suppose that $D$ is a regular language over an alphabet $A$. How can I prove that the following language is also regular? $$ \mathrm{LANGUAGE}_2(D) := \{ d \mid d,t \in A^* \text{ and } dt \in D \} $$ (This problem is taken from Introduction to the Theory of Computation by Mike Sipser.)

Answer 1

(1) If the regular language is given by a finite state automaton $M$ the language of prefixes can be obtaines by extending the set of accepting states. Choose all states that lie on a path from initial state to one of the original accepting states.

The prefix operation is a special case of the operation quotient

$K/L = \{ x \mid xy\in L, \text{ for some } y\in K\}$

where we take $L$ to be the language $A^*$ of all strings. Surprisingly, the regular languages are closed under quotient by arbitrary languages. See are regular languages closed under division, and Closure against right quotient with a fixed language.

(2) If the regular language is given by a regular expression instead, then we can construct a new regular expression for the prefix language, directly using the inductive definition of regular expressions. See Regularity of “middles” of words from regular language.

(3) Also, there is a characterization in terms of Myhill-Nerode equivalence classes: a language is regular iff its number of equivalence classes is finite. In can be observed that $x \equiv_L y$ implies $x\equiv_{\text{pref }L} y$, so if $\equiv$ is of finite index, then so is $\equiv_{\text{pref }L}$. More on this see Myhill-Nerode and closure properties.

(As Yuval noted, the exercise is somewhat standard, but I would like to have some remarks here to close the issue. Feel free to add relevant links.)