Definition:Set of Residue Classes

Definition
Let $m \in \Z$.

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

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

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

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

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