Methods for proving an equivalence relation 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.)
 A: Hint
Prove that the associated partition is 
$$\{[q]\, |\, q\in[0,1)\}$$
A: 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.
A: This post is to answer the question above:
How do I know that r∼s until I look at their difference?
Answer:
Use an indexing number m, with m = r - g(r), with g(r) being the next lower integer from r. Then each class can be represented as Am (I don't know how to subscript). m will be some rational number between zero (inclusive) and 1 (non-inclusive).
