Show that is an equivalence relation and describe the elements in the equivalence class $\operatorname{cl}(3)$.
We're studying sets and equivalence in my mathematical proofs class. As this is a proofs class I require standard proof procedure in all my answers. Don't really know how to begin this one. Help please.
To show that $\sim$ is an equivalence relation, you need to show that the three properties are verified: reflexivity, symmetry and transitivity.
Before that, note that this is an equivalence on $\mathbb{R}^*$, i.e. the set of nonzero real numbers. The definition doesn't work for $b=0$. The domain of the relation should really have been given in the problem statement. (You can pick other domains, as long as they don't contain 0. Follow-up exercises: 1. what subsets of the reals can be the domain of this relation? 2. How can you extend the definition to cope with $b=0$ so that the resulting relation is an equivalence?)
The statement of reflexivity is: for all $a \in \mathbb{R}^*$, $a/a \in H$. Given $a \in \mathbb{R}^*$, we have $a/a = 1 = 2^0 \in H$. This proves reflexivity.
I leave the proof of symmetry and transitivity to you. They can work along similar lines.
The definition of $\mathrm{cl}(3)$ is that it's the set of $x \in \mathbb{R}^*$ such that $x \sim 3$ (or the set such that $3 \sim x$, it's the same set by symmetry). That is: $$ \begin{align*} \mathrm{cl}(3) & = \{ x \in \mathbb{R}^* : x/3 \in H \} = \{ x \in \mathbb{R}^* : \exists m \in \mathbb{Z}, x/3 = 2^m \} \\ &= \{ x \in \mathbb{R}^* : \exists m \in \mathbb{Z}, x = 3 \cdot 2^m \} = \{ 3 \cdot 2^m : m \in \mathbb{Z} \} \end{align*} $$