\begin{definition}[Definition:Set of Residue Classes/Real Modulus]
Let $z \in \R$.
Let $\RR_z$ be the congruence relation modulo $z$ on the set of all $a, b \in \R$:
:$\RR_z = \set {\tuple {a, b} \in \R \times \R: \exists k \in \Z: a = b + k z}$
Let $\eqclass a z$ be the residue class of $a \pmod z$.
The quotient set of congruence modulo $z$ denoted $\R_z$ is:
:$\R_z = \dfrac \R {\RR_z}$
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 $\R$.
\end{definition}