I don’t how to prove these.

Show that the regular languages are closed under the following operations:

(a) $ $ \mathbb{DROPOUT}(L) = \{ xz \mid xyz \in L \text{ where }x,z \in \Sigma^*, y \in \Sigma \}. $ $ Namely, $ \mathbb{DROPOUT}(L)$ is the language containing all strings that can be obtained by removing one symbol from a string in $ L$ . For example, if $ L = \{012\}$ , then $ \mathbb{DROPOUT}(L) = \{12, 02, 01\}$ .

(b) $ $ \mathbb{INIT}(L) = \{ w\in \Sigma^+ \mid \text{ for some } x\in\Sigma^*,\ wx \in L\}. $ $ For example, if $ L = \{01, 110\}$ , then $ \mathbb{INIT}(L) = \{0, 01, 1, 11, 110\}$ . (HINT: Start with a DFA $ A$ for $ L$ and describe how to construct an FA for $ \mathbb{INIT}(L)$ using $ A$ . We assume that $ A$ has no sink states.)