defining a relation on the set of reals. proving that it is an equivalence relation and compelling description

Question

Define a relation $\sim$ on the set $\textbf{R}$ of the real numbers by setting $a\sim b \iff b-a \in \textbf{Z}$. Prove that this is an equivalence relation, and find a compelling description for $\textbf{R}/\sim$. Do the same for the relation $\sim$ on the plane $\textbf{R} \times \textbf{R}$ defined by declaring $(a_1, a_2)\sim (b_1,b_2) \iff b_1 - a_1 \in \textbf{Z}\land b_2 - a_2 \in \textbf{Z}.$

Answer 1

You need to show reflexivity, i.e. $a-a\in\mathbf Z$; symmetry, i.e. $a-b\in\mathbf Z\implies b-a\in\mathbf Z$; transitivity, i.e. $a-b,b-c\in\mathbf Z\implies a-c\in\mathbf Z$. Each of these is very immmediate.

Answer 2

For the "compelling description" of $\mathbb{R}/\sim$ you have the reals modulo one. So you can think of it as the interval $[0,1)$ where if you go to the right of 1, you wrap around back to zero. Or you can think of it as the unit circle $S^1$ in the complex plane where you identify $r$ with $e^{2\pi i r}$.