Quotient Set · aiwiki.page

A quotient set is the set obtained by grouping the elements of a set according to an equivalence relation and treating each group as a single element of a new set. If the original set is $X$ and the relation is $\sim$ , the quotient is written $X/{\sim}$ . Its elements are equivalence classes, not individual elements of $X$ . This construction provides the set-theoretic foundation for many quotient constructions in algebra and topology. (webhomes.maths.ed.ac.uk)

Definition and partitions

An equivalence relation on $X$ is reflexive, symmetric, and transitive. For $x\in X$ , define

[x]_{\sim}=\{y\in X:y\sim x\}.

The quotient set is

X/{\sim}=\{[x]_{\sim}:x\in X\}.

The subscript is usually omitted when the relation is clear. Since every class is a subset of $X$ , the quotient is itself a subset of the power set $\mathcal P(X)$ . A fundamental identity is

[x]=[y]\quad\Longleftrightarrow\quad x\sim y.

Thus different representatives can denote the same quotient element. (webhomes.maths.ed.ac.uk)

The classes form a partition of $X$ : they are nonempty, distinct classes are disjoint, and their union is $X$ . Conversely, a partition determines an equivalence relation by declaring two elements equivalent exactly when they lie in the same block. The quotient set is then the collection of blocks. (webhomes.maths.ed.ac.uk)

Canonical projection and universal property

The canonical projection, also called the canonical surjection, is the function

q:X\longrightarrow X/{\sim},\qquad q(x)=[x].

It is a surjective function and satisfies

q(x)=q(y)\quad\Longleftrightarrow\quad x\sim y.

Consequently, it identifies exactly those elements specified by the relation. (maths.ed.ac.uk)

Its universal property characterizes functions defined on the quotient. If $f:X\to Y$ is constant on every equivalence class—that is,

x\sim y\Longrightarrow f(x)=f(y),

then there is a unique function

\bar f:X/{\sim}\longrightarrow Y

such that

f=\bar f\circ q.

Explicitly, $\bar f([x])=f(x)$ . Constancy on classes makes this definition independent of the representative; surjectivity of $q$ gives uniqueness. This process is called factoring through the quotient or descending to the quotient. (maths.ed.ac.uk)

The universal property determines the quotient together with its projection up to a unique bijection commuting with the projections. Thus a quotient may be represented by a more convenient set rather than literally by a collection of subsets. (xenaproject.wordpress.com)

Examples

Integers modulo a positive integer

On the integers, fix $n\geq1$ and define

a\sim b\quad\Longleftrightarrow\quad n\mid(a-b).

The classes are

[a]=a+n\mathbb Z,

and the quotient has exactly $n$ elements:

\mathbb Z/n\mathbb Z=\{[0],[1],\ldots,[n-1]\}.

For example, modulo $3$ , the numbers $1$ , $4$ , and $-2$ represent the same element. This quotient underlies modular arithmetic. (maths.ed.ac.uk)

Equal outputs of a function

Every function $f:X\to Y$ induces an equivalence relation

x\sim_f x'\quad\Longleftrightarrow\quad f(x)=f(x').

The assignment $[x]\mapsto f(x)$ gives a bijection

X/{\sim_f}\longrightarrow f(X),

where $f(X)$ is the image of $f$ . In particular, every surjection exhibits its target as a model of a quotient of its domain. (maths.ed.ac.uk)

For the squaring function on the real numbers, the equivalence classes are $\{x,-x\}$ , with $\{0\}$ as a singleton class. The nonnegative real numbers provide one representative from each class. (maths.ed.ac.uk)

Representatives and well-defined constructions

A representative of a class is any member of that class. A complete set of representatives contains exactly one member of each class. It is a way to label quotient elements, not the quotient's defining collection of classes. Different choices can label the same quotient. (maths.ed.ac.uk)

A representative-based formula must give the same result for every representative. For example, a proposed rule

F([x])=g(x)

defines the intended function precisely when $x\sim y$ implies $g(x)=g(y)$ . Choosing one representative per class can produce a different, choice-dependent construction; it does not establish that the original rule is independent of representatives. (xenaproject.wordpress.com)

Likewise, a binary operation proposed by

[x]\star[y]=[x*y]

requires

x\sim x',\quad y\sim y' \quad\Longrightarrow\quad x*y\sim x'*y'.

For modular addition, this condition holds, so $[a]+[b]=[a+b]$ is well-defined. An arbitrary equivalence relation need not satisfy such compatibility conditions. (maths.ed.ac.uk)

Quotients with additional structure

A quotient set supplies the underlying elements of a structured quotient, but additional structure requires its own definition and compatibility conditions.

In group theory, the cosets of a subgroup $H$ form a quotient set $G/H$ . The usual multiplication

(gH)(kH)=gkH

makes this set a quotient group when $H$ is a normal subgroup; normality is what ensures independence from representatives. (jmilne.org)

For a ring $R$ and a two-sided ideal $I$ , the classes under $a\sim b$ when $a-b\in I$ form the quotient ring $R/I$ , with addition and multiplication induced from $R$ . (sites.millersville.edu)

In topology, a quotient space equips a quotient set with the quotient topology:

U\subseteq X/{\sim}\text{ is open} \quad\Longleftrightarrow\quad q^{-1}(U)\text{ is open in }X.

For instance, identifying the endpoints of the interval $[0,1]$ produces a space homeomorphic to a circle. The quotient set records the identification; the quotient topology supplies the topological structure. (webhomes.maths.ed.ac.uk)

References

An Introduction to Sets and Their Applicationswebhomes.maths.ed.ac.uk
MAT3-ALG algebra 2006/7 — Lecture Notesmaths.ed.ac.uk
Interlude on Setsmaths.ed.ac.uk
Formalising Mathematics: workshop 7 — quotientsxenaproject.wordpress.com
Elementary Topology: Problem Textbookwebhomes.maths.ed.ac.uk
Group Theoryjmilne.org
Quotient Ringssites.millersville.edu