Define the equivalence relation $R$ on $\mathbb{Z}$ associated with the function $f(x) = x^2 - 2x$. Write equivalence classes and quotient set.

Question

Define the equivalence relation $R$ on $\mathbb{Z}$ associated with the function $f(x) = x^2 - 2x$. Write equivalence classes and quotient set.

I guessed $x^2-2x = y^2 - 2y$ then doing some math I got $x = y$ and $y = 2 - x$. Then, I guessed the $C[1]$ is the 'rare' class, since it only has $[1]$ in its set. However, I don't know how to express the classes and the quotient set correctly, can someone give me some examples please?

Answer 1

Let us consider $x$ a fixed integer. Then $x\sim y$ if and only if $f(x)=f(y)$. If we solve this quadratic equation in terms of $y$ we get

$$y=1\pm(x-1).$$

So the equivalence classes are of the form $[y]=\{y,2-y\}$. The quotient set is naturally identified with the natural numbers $\mathbb{N}=\{1,2,\dots\}$ via the map sending $[y]$ to the unique positive integer in the equivalence class $[y]$ (Draw a picture of the equivalence classes and notice the symmetry around 1).