Definition:Residue Class

Definition
Let $z \in \Z$.

Let $\mathcal R_z$ be the congruence relation modulo $z$ on the set of all $a, b \in \Z$:


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

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

So for any $z \in \Z$, we denote the equivalence class of any $a \in \Z$ 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 $\left[\!\left[{x}\right]\!\right]_z = \left[\!\left[{y}\right]\!\right]_z \iff x \equiv y \pmod z$.

Also known as
Residue classes are sometimes known as congruence classes (modulo $z$).