This is a question from Pinter p.124. It asks me to describe the partition of the set of nonzero real numbers, $\mathbb{R}^*$ where the associated equivalence relation is $a$ ~ $b$ iff $\frac{a}{b}$ $\in \mathbb{Q}$.
I'm not sure if I can describe it as either:
{$A_r : r \in \mathbb{Q} \times \mathbb{R}^*$} where $A_r = \{x\in\mathbb{R}^* : \frac{x}{b} = a, r = (a,b)\}$
or
{$[y]:y\in \mathbb{R}^*$} where $[y] = \{ x \in\mathbb{R}^*:\frac{x}{y} \in \mathbb{Q}\}$
assuming that each disjoint set can have repeating labels? Thanks a lot.
Hint 1: Given a rational number $q$, for every non-zero rational $p$, $p/q$ is also rational. Furthermore, for an irrational $x$, $q/x$ is irrational. This argument classifies the equivalence classes whose elements are all rational.
Hint 2: For the equivalence classes containing irrational numbers, take an irrational $x$. Then, if $x \sim y$ for $y \in \mathbb{R}^*$, then $y$ must be irrational (why?). Recall that $[x]$ contains all $y$ such that $x \sim y$, so determining the elements of $[x]$ comes down to determining when the ratio $x/y$ of two irrational numbers is rational.