Equivalence Class/Examples/Congruence Modulo Initial Segment of Natural Numbers/Examples/4
Jump to navigation
Jump to search
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 $\RR_m$:
- $\eqclass a m := \set {a + z m: z \in \Z}$
Then:
\(\ds \eqclass 3 4\) | \(=\) | \(\ds \set {\dotsc, -13, -9, -5, -1, 3, 7, 11, 15, \dotsc}\) |
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