I'm having a little trouble wrapping my head around the elements of the equivalence classes using the following definition:
for m, n in N, define m ~ n if m^2 - n^2 is a multiple of 3.
1) List four elements in the equivalence class of [0].
2) List four elements in the equivalence class of [1].
3) Do you think there are any more equivalence classes? Actually provide a proof.
I'm having trouble figuring out what's in each equivalence class. Also, I believe that there are 3 total equivalence classes from looking at the possible remainders when dividing by 3 (0, 1, 2). How do I determine the elements in each equivalence class?
Do I just set m to 0 and see what values for n are multiples of 3 to determine the members of [0]? Similarly, do I just set m to 1 for [1] and m to 2 for [2]?
0^2 - 3^2 = -6 which is a multiple of 3. (Since m,n are in N, is it fine that -6 was the result?)
1^2 - 2^2 = -3. This actually confuses me a little. Wouldn't this mean that the equivalence class for [2] is the same as the equivalence class for [1]? If so, there would only be 2 equivalence classes, and I'm not quite sure how I would go about proving that.
Edit: Using the above, I'm beginning to think there are only 2 equivalence classes since [2] is actually the same as [1]. However, I'm not quite sure how to prove this. Would the proof go something along the lines of using the division algorithm to show that there are only 3 possible remainders. Then to show that the equivalence class for [2] is actually the same one for [1]?
Yes, you need to unwrap the definition: it means you are putting numbers in the equivalence classes of residues mod 3 of their squares. Consider three cases:
3n : class is (3n)2 mod 3 = [0]
3n+1 : class is (3n+1)2 mod 3 = [1]
3n+2 : class is (3n+2)2 mod 3 = [1]
So there are only two classes. This seems to be an exhaustive demonstration, but no doubt a polished "proof" would use slightly different words.