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 residue classes can also be seen as the complete set of residues.

Some sources prefer the term  set of all residue classes but it is 's opinion that the all is redundant.

Also see

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


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