$4.$ For a language L CE$*,$ define ins$($L$)$ to be the language containing all strings that can be obtained by taking a string in L and inserting one character from E in the second position in the string. That is$,$ ins$($L$)=\{($w$)/($w$)=$ xat, where x e E$,$ a $$ Ente $*,$ and $$t e L$\}$ Prove that the class of regular languages is closed under the ins operation. In particular, given a DFA recognizing L$,$ show how to construct an NFA that recognizes ins$($L$).$ Justify your construction.

For a language Lsube$\backslash$Sigma $^(*),$ define ins$($L$)$ to be the language containing all strings that can

be obtained by taking a string in L and inserting one character from $\backslash$Sigma in the second

position in the string. That is$,$

ins$($L$)=\{$w$|$w$=$xat, where xin$\backslash$Sigma $,$ain$\backslash$Sigma $,$tin$\backslash$Sigma $^(*),$ and xtinL$\}$

Prove that the class of regular languages is closed under the ins operation.

In particular, given a DFA recognizing L$,$ show how to construct an NFA that recognizes

ins$($L$).$ Justify your construction.