Quotient Mapping/Examples

Examples of Quotient Mappings

Congruence Modulo $3$

Let $x \mathrel \RR y$ be the equivalence relation defined on the integers as congruence modulo $3$:

$x \mathrel \RR y \iff x \equiv y \pmod 3$

defined as:

$\forall x, y \in \Z: x \equiv y \pmod 3 \iff \exists k \in \Z: x - y = 3 k$

That is, if their difference $x - y$ is a multiple of $3$.

From Congruence Modulo $3$, the quotient set induced by $\RR$ is:

$\Z / \RR = \set {\eqclass 0 3, \eqclass 1 3, \eqclass 2 3}$

Hence the quotient mapping $q_\RR: \Z \to \Z / \RR$ is defined as:

$\forall x \in \Z: \map {q_\RR} x = \eqclass x 3 = \set {x + 3 k: k \in \Z}$

Modulo $2 \pi$ as Angular Measurement

Let $\RR$ denote the congruence relation modulo $2 \pi$ on the real numbers $\R$ defined as:

$\forall x, y \in \R: \tuple {x, y} \in \RR \iff \text {$x$ and $y$}$ measure the same angle in radians

From Congruence Modulo $2 \pi$ as Angular Measurement, the quotient set induced by $\RR$ is:

$\R / \RR = \set {\eqclass \theta {2 \pi}: 0 \le \theta < 2 \pi}$

where:

$\eqclass \theta {2 \pi} = \set {\theta + 2 k \pi: k \in \Z}$

Hence the quotient mapping $q_\RR: \R \to \R / \RR$ is defined as:

$\forall x \in \R: \map {q_\RR} x = \eqclass x {2 \pi} = \set {x + 2 k \pi: k \in \Z}$