# languages are closed under the following operations

I don't how to prove these.

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

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