Definition:Set of Residue Classes
Definition
Let $m \in \Z$.
Let $\RR_m$ be the congruence relation modulo $m$ on the set of all $a, b \in \Z$:
- $\RR_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$
Let $\eqclass a m$ be the residue class of $a$ (modulo $m$).
The quotient set of congruence modulo $m$ denoted $\Z_m$ is:
- $\Z_m = \dfrac \Z {\RR_m}$
Least Positive Residues
Let $\eqclass a m$ be the residue class of $a$ (modulo $m$).
Let $r$ be the smallest non-negative integer in $\eqclass a m$.
Then from Integer is Congruent to Integer less than Modulus:
- $0 \le r < m$
and:
- $a \equiv r \pmod m$
Then $r$ is called the least positive residue of $a \pmod m$.
Least Absolute Residues
Let $\eqclass a m$ be the residue class of $a$ (modulo $m$).
Except when $r = \dfrac m 2$, we can choose $r$ to be the integer in $\eqclass a m$ which has the smallest absolute value.
In that exceptional case we have:
- $-\dfrac m 2 + m = \dfrac m 2$
and so:
- $-\dfrac m 2 \equiv \dfrac m 2 \pmod m$
Thus $r$ is defined as the least absolute residue of $a$ (modulo $m$) if and only if:
- $-\dfrac m 2 < r \le \dfrac m 2$
Real Modulus
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 $\R$.
Also known as
The set of residue classes can also be seen as the complete set of residues or complete residue system.
Some sources prefer the term set of all residue classes but it is $\mathsf{Pr} \infty \mathsf{fWiki}$'s opinion that the all is redundant.
Examples
Set of Residue Classes Modulo $2$
The elements of $\Z_2$, the set of residue classes modulo $2$, are:
\(\ds \eqclass 0 2\) | \(=\) | \(\ds \set {\dotsc, -6, -4, -2, 0, 2, 4, 6, \dotsc}\) | ||||||||||||
\(\ds \eqclass 1 2\) | \(=\) | \(\ds \set {\dotsc, -5, -3, -1, 1, 3, 5, \dotsc}\) |
Set of Residue Classes Modulo $3$
The elements of $\Z_3$, the set of residue classes modulo $3$, are:
\(\ds \eqclass 0 3\) | \(=\) | \(\ds \set {\dotsc, -6, -3, 0, 3, 6, \dotsc}\) | ||||||||||||
\(\ds \eqclass 1 3\) | \(=\) | \(\ds \set {\dotsc, -5, -2, 1, 4, 7, \dotsc}\) | ||||||||||||
\(\ds \eqclass 2 3\) | \(=\) | \(\ds \set {\dotsc, -4, -1, 2, 5, 6, \dotsc}\) |
Set of Residue Classes Modulo $4$
The elements of $\Z_4$, the set of residue classes modulo $4$, are:
\(\ds \eqclass 0 4\) | \(=\) | \(\ds \set {\dotsc, -8, -4, 0, 4, 8, 12, 16, \dotsc}\) | ||||||||||||
\(\ds \eqclass 1 4\) | \(=\) | \(\ds \set {\dotsc, -7, -3, 1, 5, 9, 13, 17, \dotsc}\) | ||||||||||||
\(\ds \eqclass 2 4\) | \(=\) | \(\ds \set {\dotsc, -6, -2, 2, 6, 10, 14, 18, \dotsc}\) | ||||||||||||
\(\ds \eqclass 3 4\) | \(=\) | \(\ds \set {\dotsc, -5, -1, 3, 7, 11, 15, 19, \dotsc}\) |
Set of Residue Classes Modulo $5$
The elements of $\Z_5$, the set of residue classes modulo $5$, are:
\(\ds \eqclass 0 5\) | \(=\) | \(\ds \set {\dotsc, -10, -5, 0, 5, 10, 15, 20, \dotsc}\) | ||||||||||||
\(\ds \eqclass 1 5\) | \(=\) | \(\ds \set {\dotsc, -9, -4, 1, 6, 11, 16, 21, \dotsc}\) | ||||||||||||
\(\ds \eqclass 2 5\) | \(=\) | \(\ds \set {\dotsc, -8, -3, 2, 7, 12, 17, 22, \dotsc}\) | ||||||||||||
\(\ds \eqclass 3 5\) | \(=\) | \(\ds \set {\dotsc, -7, -2, 3, 8, 13, 18, 23, \dotsc}\) | ||||||||||||
\(\ds \eqclass 4 5\) | \(=\) | \(\ds \set {\dotsc, -6, -2, 1, 9, 14, 19, 24, \dotsc}\) |
Also see
- Results about residue classes can be found here.
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $4$
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)}$
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Theorem $5$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Equivalence Relations: $\S 18$
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.3$: Equivalence Relations: Problem Set $\text{A}.3$: $18$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 18$: Congruence classes
- 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: $1.5$ Relations: Equivalence Relations: Example $1.4$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.8$ Algebraic properties of $p$-adic integers