# Definition:Set of Residue Classes/Real Modulus

## Definition

Let $z \in \R$.

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

$\mathcal R_z = \set {\tuple {a, b} \in \R \times \R: \exists k \in \Z: a = b + k z}$

Let $\eqclass a z$ be the residue class of $a \pmod z$.

The quotient set of congruence modulo $z$ 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 $\R$.

## Also known as

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