Construct Turing machines that compute the following number$-$theoretic functions and relations. Do not use macros in the design of these machines.

e$)eq(n,m)=\{\begin{array}{ccc}1& if& n=m\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$