Let $H = \{2^m : m \in \mathbb{Z}\}$ & define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rationals by $a\mathbin{R}b$ iff $a/b \in H$.

Question

Let $H = \{2^m : m \in \mathbb{Z}\}$ and define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rational numbers by $a\mathbin{R}b$ if and only if $a/b \in H$. Prove that $R$ is an equivalence relation on $\mathbb{Q^{+}}$.

So far I have:

Let $x,y \in \mathbb{Q^{+}}$. Then $x\mathbin{R}y$ if and only if $\frac{x}{y} \in H$. Then $\frac{x}{y} = 2^{m}$ for some $m \in \mathbb{Z}$. Then $\frac{x}{x} = 1$. Then, $\frac{x}{y} = 2^{m}$ for some $m \in \mathbb{Z}$. Hence, $\frac{y}{x} = 2^{-m}$.

I'm not sure what to do next or how to show that $R$ is an equivalence relation on $\mathbb{Q^{+}}$.

Answer 1

You have done most of the work, you just need to prove transitivity.

If $a R b$ and $b R c$, $\frac{a}{b}=2^{m_1}$ and $\frac{b}{c}=2^{m_2}$,

multiply them together, you have

$$\frac{a}{c}=2^{m_1+m_2}$$

Since $m_1+m_2 \in \mathbb{Z}$, $\frac{a}{c} \in H$, i.e. $a Rc$.