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.

$\blacksquare$

Examples

Equivalence Class for Initial Segment of Natural Numbers Modulo $4$

\(\ds \eqclass 3 4\)

\(=\)

\(\ds \set {\dotsc, -13, -9, -5, -1, 3, 7, 11, 15, \dotsc}\)

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets