- #1

frankfjf

- 168

- 0

## Homework Statement

Determine if this is an equivalence relation. Either specify which properties fail or list the equivalence classes:

A = {0, 1, 2...}

R = {(m,n) | m^2 ≡ n^2 mod 3}

## Homework Equations

m^2 ≡ n^2 mod 3

## The Attempt at a Solution

I've determined that it is indeed an equivalence relation, but my problem is when it comes to coming up with the equivalence classes. I'm used to the equation in the relation just using m and n instead of them being squared, so perhaps that's what's throwing me off.

[0] = {0, 3, 6, 9, ...} but unlike equivalence relations where [1] would be {1, 4, 7, 10, ...} the professor's solution says that [1] = {1, 2, 4, 5, 7, 8, ...}. Why is this? What happens in this particular equivalence relation that causes [1] to have that pattern? I know that you have to look at it as m^2 - n^2 = 3z, but how does the squaring change the pattern?