showing a relation on $\mathbb Z$ \ $0$ is an equivalence relation We  define  a  relation  on $\mathbb Z \setminus {0}$ where a ~ b iff   $0< ab$.  How would you show  this  is  an  equivalence relation and describe the equivalence classes?

I'm using the notation $a \simeq b$ for the relation since I can't get the $\LaTeX$ for the "tilde" sign to work right at the moment.
$ab > 0 \tag 1$
if and only if the signs of $a$ and $b$ are the same; thus
$a \simeq a, \tag 1$
since $a$ has the same sign as itself;
$a \simeq b \Longrightarrow b \simeq a, \tag 2$
since if $a$ has the same sign as $b$, then $b$ has the same sign as $a$;
$[a \simeq b] \wedge [b \simeq c] \Longrightarrow [a \simeq c], \tag 3$
since if $a$ has the same sign as $b$ and $b$ has the same sign as $c$ . . . well, you get the idea . . . 
