The relation $\sim$ in $\mathbb{R}$ is defined as: $x \sim y \iff x − y \in \mathbb{Z}$ $(a) \quad$ Show that $\sim$ is an equivalent relation.
$(b) \quad$ Give $2$ distinct equivalent classes (must show they are distinct).
$(c) \quad$ $[0, 1) = \{x \in \mathbb{R} \mid 0 \leq x < 1\}.$ Show that $^{\mathbb{R}}\big/_{\sim} \cong [0, 1).$
I was able to prove that the relation is reflexive, symmetric and transitive. So, it is equivalent.
For $(b)$ one class I could find was
$$\{y \in \mathbb{R} \mid x \sim y\} = \{y \in \mathbb{R} \mid \exists z \in \mathbb{Z}, x - y = z\} = \{y \in \mathbb{R} \mid \exists z \in \mathbb{Z}, x = z + y\}.$$
But I cannot find another one which is distinct!
And for $(c)$ I don't even know how to start this proof. Any ideas? Thanks

(a) It's called equivalence relation
(b) The set you provided here is the equivalence class of a fixed element $x$. For example what is the equivalence class of $0$, or of $1/2$?
(c) Consider a specific function $f:\Bbb R\to [0,1)$ which satisfies $f(x)=f(y)\iff x\sim y$.
