# Equivalence Relation and functions question

Could someone please confirm if I understand this correctly? Here is the problem:

define ~ on Z by m ~ n in case m^2 ~ n^2.

1) What, if anything, is wrong with the following "definition" of a function f: [Z] -> Z? Let f([m]) = m^2 + m + 1.

2) What, if anything, is wrong with the following "definition" of the operation (+) on [Z]? Let [m] (+) [n] = [m + n].

Am I correct in saying that the equivalence classes are 1, 2, 3, ... n? Since m^2 = n^2 only if m and n are the same or are the negative versions of themselves? For example, the first two equivalence classes are [1] and [2]. 1's members are {1, -1} and 2's members are {2, -2}.

For the first one, is the following logic correct? One equivalence class in Z is 1, so I plug 1 into the function to get 1^2 + 1 + 1 = 3. An equivalent element in the same equivalence class is -1. I plug -1 in to get (-1)^2 + (-1) + 1 = 1. Since these two elements in the same equivalence class do NOT map to equivalent elements (3 =/= 1), there is a problem with the function.

For the second function, [1] (+) [2] = [1 + 2] = [3] and [-1] (+) [-2] = [-1 - 2] = [-3]

There is no problem since the equivalence class for 3 is the same one as -3.

If any of this is incorrect, could you please explain why and point me in the right direction? Thank you very much

In 1), your function is not well defined since if $m \neq n$ and $m, n \in \mathbb{Z}$, then we don't necessarily have that $f([m]) = f([n])$ even if $[m] = [n]$.