Definition:Set of Residue Classes

Definition
Let $m \in \Z$.

Let $\mathcal R_m$ be the congruence relation modulo $m$ on the set of all $a, b \in \Z$:
 * $\mathcal R_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$

Let $\eqclass a m$ be the residue class of $a$ (modulo $m$).

The quotient set of congruence modulo $m$ denoted $\Z_m$ is:
 * $\Z_m = \dfrac \Z {\mathcal R_m}$

Also known as
The set of all residue classes can also be seen as the complete set of residues.

Also see

 * Congruent Integers in Same Residue Class
 * Residue Classes form Partition of Integers


 * Definition:Integers Modulo m
 * Definition:Zero Residue Class