For any language $E,$ we can define another language Nearly$(E)$ which contains all strings that

differ from some string in $E$ by exactly one letter. For example, if $=\{a,b\}$ and $E=\{aa,ab,$aaa$\}$

then Nearly $(E)=\{ab,ba,aa,bb,$aab,aba,baa$\}.$ Note that to generate new strings in Nearly $(E),$

we change characters in strings from $E,$ but we do not remove or add letters. Describe how to take

a DFA for $E$ and construct an NFA for Nearly $(E).$ Your description should have five parts:

A high$-$level intuitive description of the procedure for constructing the NFA.

An example where you build the construction for an example DFA.

A formal description of the new NFA, based on the DFA $M=(Q,,,s,F).$

Given a string in Nearly $(E),$ how do you construct an accepting path in your automaton

$($using knowledge about the location of the typo and the accepting path in $M).$

Given a string your automaton accepts, how do you use its accepting path to convince the

reader that it is in Nearly $(E)?$

Note that this is not asking for a fully formal proof of correctness, with lots of lines of symbols and

$>s.$ However, the above steps contain all the ideas necessary to produce such a thing if you really

had to$.$

Hint: It will be helpful to make two copies of the DFA for $E.$