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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

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1 Answer 1

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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 $[m] = [n]$.

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