Definition:Residue Class

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Definition

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let $\RR_m$ be the congruence relation modulo $m$ on the set of all $a, b \in \Z$:

$\RR_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$


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

So for any $m \in \Z$, we denote the equivalence class of any $a \in \Z$ by $\eqclass a m$, such that:

\(\ds \eqclass a m\) \(=\) \(\ds \set {x \in \Z: a \equiv x \pmod m}\)
\(\ds \) \(=\) \(\ds \set {x \in \Z: \exists k \in \Z: x = a + k m}\)
\(\ds \) \(=\) \(\ds \set {\ldots, a - 2 m, a - m, a, a + m, a + 2 m, \ldots}\)


The equivalence class $\eqclass a m$ is called the residue class of $a$ (modulo $m$).


Examples

Modulo $7$

The residue class of $2$ modulo $7$ on the integers is:

$\eqclass 2 7 = \set {\ldots, -19, -12, -5, 2, 9, 16, \ldots}$


Also defined as

In their definition of a residue class, some sources are lax at defining $m$ as a strictly positive integer, and the restriction becomes clear only during further development of the theory.

Some sources with particular aims in mind are deliberately explicit about specifying that $m > 1$.


Also known as

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


Also see

  • Results about residue classes can be found here.


Sources

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