= Quotient set
{title2=$X/{\sim}$}
{wiki}
A quotient set packages each <equivalence class> as a single new <set element>. It remembers which class an original <set element> belongs to and forgets which member of that class it was. For an <equivalence relation> $\sim$ on a <set> $X$,
$$
X/{\sim}=\{[x]:x\in X\}.
$$
Each <set element> of the quotient set is itself a nonempty <subset> of $X$. The <canonical projection of a quotient set> sends an original <set element> to this entire <equivalence class>. https://www.math.umd.edu/~pbrosnan/notes/ugalg/sect0018.html[Patrick Brosnan, Undergraduate Algebra, §2.10, Definitions 2.78 and 2.80]{ref}
For <integer congruence> modulo three restricted to $X=\{0,1,\ldots,8\}$, the quotient set is
$$
X/{\sim}=\bigl\{\{0,3,6\},\{1,4,7\},\{2,5,8\}\bigr\}
=\{[0],[1],[2]\}.
$$
It has <cardinality> three. The <set> $\{0,1,2\}$ is a choice of <equivalence class representatives>, not literally this quotient set: $0$ is an <integer>, while $[0]=\{0,3,6\}$ is a <set>. The <function> sending each chosen <equivalence class representative> to its <equivalence class> is a <bijection> between these two different <sets>.
\Image[/quotient-map-collapsing-classes.png]
{title=The quotient projection sends nine elements to three classes, with each class appearing as one fiber}
A quotient set alone specifies no arithmetic. To obtain the familiar <modular arithmetic> example with <addition>, use <integer congruence> on the whole <set> of <integers>. Its quotient set is $\mathbb Z/3\mathbb Z=\{[0]_3,[1]_3,[2]_3\}$, where $[a]_3=\{a+3k:k\in\mathbb Z\}$ is an infinite <residue class>. Define <addition> by
$$
[a]_3+[b]_3=[a+b]_3.
$$
The answer must be independent of the <equivalence class representatives>. If $a'=a+3k$ and $b'=b+3\ell$, then $a'+b'=a+b+3(k+\ell)$, so both sums have the same <residue class>. This is what it means for the <group operation> to be well-defined. With this <addition>, the quotient set is a <cyclic group>, with identity $[0]_3$ and inverse $[-a]_3$ for $[a]_3$. https://math-cs.gordon.edu/ntic/ntic2017/section-cong-props.html[Karl-Dieter Crisman, Number Theory: In Context and Interactive, §4.3, Proposition 4.3.2: congruence arithmetic is well-defined]{ref}
For example, $[2]_3+[2]_3=[4]_3=[1]_3$. Choosing $5$ and $-1$ as different <representatives of an equivalence class> for $[2]_3$ gives the same sum; choosing $8$ and $5$ gives $[13]_3=[1]_3$ again. These calculations use <residue classes> of all <integers>. Raw <addition> on the finite $X=\{0,\ldots,8\}$ does not stay in $X$: $8+8=16\notin X$. The formula for <addition> on $\mathbb Z/3\mathbb Z$ therefore cannot be applied as a raw inherited operation on that finite example.
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