Equivalence Class/Examples/Congruence Modulo Initial Segment of Natural Numbers

Example of Equivalence Class
Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let $\N_{<m}$ denote the initial segment of the natural numbers $\N$:
 * $\N_{<m} = \set {0, 1, \ldots, m - 1}$

Let $\RR_m$ denote the equivalence relation:
 * $\forall x, y \in \Z: x \mathrel {\RR_m} y \iff \exists k \in \Z: x = y + k m$

For each $a \in \N_{<m}$, let $\eqclass a m$ be the equivalence class of $a \in \N_{<m}$ under $\RR_m$ is the set:
 * $\eqclass a m := \set {a + z m: z \in \Z}$

Proof
From Congruence Modulo Integer is Equivalence Relation, $\RR_m$ is an equivalence relation.

From the Division Theorem:
 * $\forall n \in \Z: \exists! z, a \in \Z: n = a + z m, a \in \N_{<m}$

Hence the result.