Definition:Residue Class

Definition
Let $$z \in \R$$.

Let $$\mathcal{R}_z$$ be the congruence relation modulo $z$ on the set of all $$a, b \in \R$$:


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

We have that congruence modulo $z$ is an equivalence relation.

So for any $$z \in \R$$, we denote the equivalence class of any $$a \in \R$$ by $$\left[\!\left[{a}\right]\!\right]_z$$, such that:

$$ $$ $$

The equivalence class $$\left[\!\left[{a}\right]\!\right]_z$$ is called the residue class of $$a$$ (modulo $$z$$).

It follows directly from the definition of equivalence class that $$\left[\!\left[{x}\right]\!\right]_z = \left[\!\left[{y}\right]\!\right]_z \iff x \equiv y \left({\bmod\, z}\right)$$.

These residue classes are known as congruence classes.

Set of All Residue Classes
The quotient set of congruence modulo $z$ is denoted $$\R_z$$ is:
 * $$\R_z = \frac {\R} {\mathcal{R}_z}$$

Thus $$\R_z$$ is the set of all residue classes modulo $$z$$.

It follows from the Fundamental Theorem on Equivalence Relations that the quotient set $$\R_z$$ of congruence modulo $z$ forms a partition of $$\R$$.

See Integers Modulo m for an application of this concept to the domain of integers.