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:
 * $\R_z = \dfrac \R {\mathcal R_z}$

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.