$(a,b)R(x,y) \iff ay=bx$ is an equivalence on $\Bbb Z \times (\Bbb Z\setminus\{0\})$

Question

Define a relation $R$ on $\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$ by $(a,b)R(x,y)\iff ay=bx.$

$a)$ Prove that $R$ is an equivalence relation.

$b)$ Describe the equivalence classes corresponding to $R$.

I know that an equivalence relation satisfies transitivity, reflexivity, and symmetry. Also, I know that the Cartesian Product is $A\times B=\{(a,b):a\in A$ and $b\in B\}. $If I could get a hint for $a)$, that would be appreciated. For part $b)$, could someone explain what an equivalence class is? Please refrain from complete solutions. Thank you.

Answer 1

For a, you need to show all three elements of the definition are satisfied. So show that $(a,b)R(a,b)$, $(a,b)R(c,d) \implies (c,d)R(a,b)$ and transitivity. Go back to the definition of $R$ and they should fall out.

Answer 2

Two elements $(a,b)$ and $(x,y)$ are in the same equivalence class iff they are related to each other. In this case, it happens iff $$ \frac{a}{b} = \frac{x}{y} $$ ie. They define the same rational number (For instance $(1,2) \sim (2,4)$.

Answer 3

For (a), you really just need to go through each property.

e.g. to check if it's reflexive, is $(a,b)\text{~}(a,b)$ for all $(a,b)\in \mathbb{Z} \times\mathbb{Z} \backslash \{0\}$?

For (b), the equivalence class of an element is the set of all elements that are related to that element.