Is $\mathbb R/R$ a partition of $\mathbb R$ given by some equivalence relation $R$?

Question

Let $aRb $ iff $b - a$ is an integer.

$5 - 0$ is an integer, so $5 \in [0].$ In fact, $[0] = \mathbb Z$. Does it mean $\mathbb Z \in \mathbb R/R$?

$5.14159265359 - \pi$ is an integer, so $5.14159265359 \in [\pi]$ and $[\pi] \in \mathbb R/R$? Is that right?

Can we list all/most/main elements of $\mathbb R/R$ in this manner?

Answer 1

Yes, $\mathbb{Z} \in \mathbb{R} / R$, and $[\pi] \in \mathbb{R}/R$. Regarding the last question, $$\mathbb{R}/R = \{[x] \mid 0 \leq x < 1 \},$$ and for $0 \leq x,y < 1$ you have $[x] = [y]$ if and only if $x = y$, so this gives you a way to think about all the elements of $\mathbb{R}/R$.