Equivalence Class for Abstract Algebra Class Let
$$R_3= \{(a,b)\mid a,b \in \mathbb{Z}\text{ and there exists }k \in \mathbb{Z} \text{ such that }a-b=3k\}.$$
I know there is an equivalence relation but I'm not 100% on what it means to be an equivalence class for this problem. In class we got 3: $\{0,3,6,9,\ldots\}$ and $\{1,4,7,10,-2,-5,\ldots\}$ and $\{2, 5, 8, 11, -1, -4,\ldots\}$. 
I don't understand where these cells came from. Help?

An exercise in set theory (if $k$ runs through $\mathbb Z$, then so does $-k$):
$[a] = \\
\{b \in \mathbb{Z} \text{ such that there exists }k \in \mathbb{Z} \text{ such that }a-b=3k\}=\\
\{b \in \mathbb{Z} \text{ such that there exists }k \in \mathbb{Z} \text{ such that }b=a-3k\}=\\
\{b \in \mathbb{Z} \text{ such that there exists }k \in \mathbb{Z} \text{ such that }b=a+3(-k)\}=\\
\{a+3k | k \in \mathbb{Z}\}=\\
\{\dots,a-6,a-3,a,a+3,a+6,\dots\}$
You should have written
$
\{\ldots, -5, -2, 1, 4, 7, 10,\ldots\}
$ instead of
$
\{1,4,7,10,-2,-5,\ldots\}
$
