Definition:Quotient Mapping

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Let $\RR \subseteq S \times S$ be an equivalence on a set $S$.

Let $\eqclass s \RR$ be the $\RR$-equivalence class of $s$.

Let $S / \RR$ be the quotient set of $S$ determined by $\RR$.

Then $q_\RR: S \to S / \RR$ is the quotient mapping induced by $\RR$, and is defined as:

$q_\RR: S \to S / \RR: \map {q_\RR} s = \eqclass s \RR$

Effectively, we are defining a mapping on $S$ by assigning each element $s \in S$ to its equivalence class $\eqclass s \RR$.

If the equivalence $\RR$ is understood, $\map {q_\RR} s$ can be written $\map q s$.

Also known as

The quotient mapping is often referred to as:

the canonical surjection from $S$ to $S / \RR$
the canonical map or canonical projection from $S$ onto $S / \RR$
the natural mapping from $S$ to $S / \RR$
the natural surjection from $S$ to $S / \RR$
the classifying map or classifying mapping (as it classifies the elements of $S$ into those various equivalence classes)
the projection from $S$ to $S / \RR$

Some sources denote the quotient mapping by $\natural_\RR$. This is logical, as $\natural$ is the "natural" sign in music.

Some sources use $\pi$ to denote the quotient mapping.


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:

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


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

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

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

Also see

  • Results about quotient mappings can be found here.

Linguistic Note

The word quotient derives from the Latin word meaning how often.