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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left[\!\left[{a}\right]\!\right]_z\)	\(=\)	\(\displaystyle \left\{ {x \in \R: a \equiv x \left({\bmod\, z}\right)}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\{ {x \in \R: \exists k \in \Z: x = a + k z}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\{ {\ldots, a - 2z, a-z, a, a+z, a+2z, \ldots}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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.

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$: Example $4$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 2.5$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.6$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 18$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 6$: Example $6.8$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 18$
• John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Example $2.30$