showing a relation on $\mathbb Z$ \ $0$ is an equivalence relation

Question

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?

Answer 1

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 . . .