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 $a, b \in \Z$:
 * $\mathcal R_m = \left\{{\left({a, b}\right) \in \Z \times \Z: \exists k \in \Z: a = b + k m}\right\}$

Let $\left[\!\left[{a}\right]\!\right]_m$ be the residue class of $a$ (modulo $m$).

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

Thus $\Z_m$ is the set of all residue classes modulo $m$:
 * $\Z_m = \left\{ {\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m - 1}\right]\!\right]_m}\right\}$

It follows from the Fundamental Theorem on Equivalence Relations that the quotient set $\Z_m$ of congruence modulo $m$ forms a partition of $\Z$.

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

Also see

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