$[10$ points$]$ Given some alphabet $,$ we define the dist operator between two

strings $x,$yin${}^{**}$ as the number of positions in which the strings $x$ and $y$

differ. That is$,$

dist$(x,y)=\{\begin{array}{ccc}n& if|x|=|y|\text{a}\text{n}\text{d}\text{t}\text{h}\text{e}\text{y}\text{d}\text{i}\text{f}\text{f}\text{e}\text{r}in\text{e}\text{x}\text{a}\text{c}\text{t}\text{l}\text{y}n\text{p}\text{o}\text{s}\text{i}\text{t}\text{i}\text{o}\text{n}\text{s}& \\ & if& xy\end{array}$

For instance dist$(aa,$baa$)=,$dist$(aab,$baa$)=2,$dist$(aba,$bab$)=3.$

Claim: The language dist$\_2$ s$.$t$.$ dist $(x,w)2$

is regular.