Methods for proving an equivalence relation

Question

I'll be taking introductory abstract algebra in the fall, and so to prepare, I'm working through Pinter's text. Chapter 12 includes a number of exercises asking the student to prove that something is an equivalence relation and to describe the associated partition. For example: In $\mathbb{Q}$, $r \sim s$ iff $r - s \in \mathbb{Z}$.

I think I'm okay with most (?) of this. I'd show this is an equivalence relation like so: If $x, y, z \in \mathbb{Q}$, then $x - x = 0 \in \mathbb{Z}$, and so $x \sim x$. Second, if $x \sim y$, then $y - x = -(x - y) \in \mathbb{Z}$, and so $x \sim y \implies y \sim x$. Finally, if $x \sim y$ and $y \sim z$, then $x - z = (x - y) - (z - y) \in \mathbb{Z}$, and so $x \sim z$.

The two parts I'm not sure about: First, that last f on the iff. Essentially, this means I have to prove that if $r \sim s$, then their difference is an integer, right? But I thought this is merely how this particular equivalence relation is defined. How do I know that $r \sim s$ until I look at their difference?

Second, I have a basic idea of what the partition is, but I'm not sure how to form the statement. The equivalence class $[q] = \{k + q : k \in \mathbb{Z}, q \in \mathbb{Q} \}$. But if anything, that seems as though it would be the definition for a single equivalence class, not the description of the partition. (Now that I look at it again, it also leaves out the fact that $k$ is arbitrary but $q$ is fixed.)

If anyone can offer any hints, I'd very much appreciate it. (Though I'm guessing the second of my questions might be more amenable to a No-this-is-how-you-do-it than to a hint, per se.)

Answer 1

Hint

Prove that the associated partition is $$\{[q]\, |\, q\in[0,1)\}$$

Answer 2

For your first part: Definitions are iff, or if and only if statements, as they are essentially stating that two things are equivalent -- the new term, and its definition. It shouldn't be affecting what you need to prove, except that you can use both that $r \sim s \implies r - s \in \mathbb{Z}$ and that $r - s \in \mathbb{Z} \implies r \sim s$.

For the second part, a better way of putting it would be as follows:

For each $q \in \mathbb{Q}$, the equivalence class of $q$ under $\sim$ is as follows: $[q] = \{q + k : k \in \mathbb{Z}\}$.

By saying this, you describe all of the equivalence classes making up the partition in one statement, and also take care of having $q$ fixed and $k$ varying.