In order to consider sign integer (both positive and negative) over some integer modulo space, should I use symmetric interval?

Question

Can I use minus operation over mod 5, i.e. {0,1,2,3,4}?? If it was {-2 -1 0 1 2} I understand that works fine since we potentially have negative element.

Answer 1

You must regard "modulo 5" as a set of equivalence classes of integers. Any equivalence class is uniquely determined by any of its members. For example, the equivalence class $\;\{5n+4: n\in\mathbb{Z}\}\;$ could be identified by either $4$ or $-1$ or any other of its members. Thus, the entire set of residues "modulo 5" can be determined by $\;\{0,1,2,3,4\}\;$ or $\;\{-2,-1,0,1,2\}\;$ or any set of five integers that are in different equivalence classes. To be pedantic, you would use notation something like $\;[4]\;$ to denote the equivalance class which contains $4$ in order to avoid confusing the class with any of its members. For example, $\;[4]=[-1]\;$ while $\;4\ne-1.\;$ Another notation is $\;4\equiv-1 \pmod 5.\;$