Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Quotient set"

(New page: A '''quotient set''' is a set derived from another by an equivalence relation. Let <math>S</math> be a set, and let <math>\mathcal{R}</math> be an equivalence relation. The set o...)
 
(added note on saturation)
Line 2: Line 2:
  
 
Let <math>S</math> be a set, and let <math>\mathcal{R}</math> be an equivalence relation.  The set of [[equivalence class]]es of <math>S</math> with respect to <math>\mathcal{R}</math> is called the ''quotient of <math>S</math> by <math>\mathcal{R}</math>'', and is denoted <math>S/\mathcal{R}</math>.
 
Let <math>S</math> be a set, and let <math>\mathcal{R}</math> be an equivalence relation.  The set of [[equivalence class]]es of <math>S</math> with respect to <math>\mathcal{R}</math> is called the ''quotient of <math>S</math> by <math>\mathcal{R}</math>'', and is denoted <math>S/\mathcal{R}</math>.
 +
 +
A [[subset]] <math>A</math> of <math>S</math> is said to be ''saturated'' with respect to <math>\mathcal{R}</math> if for all <math>x,y \in S</math>, <math>x\in A</math> and <math>\mathcal{R}(x,y)</math> imply <math>y\in A</math>.  Equivalently, <math>A</math> is saturated if it is the union of a family of equivalence classes with respect to <math>\mathcal{R}</math>.  The ''saturation of <math>A</math> with respect to <math>\mathcal{R}</math>'' is the least saturated subset <math>A'</math> of <math>S</math> that contains <math>A</math>.
  
 
== Compatible relations; derived relations; quotient structure ==
 
== Compatible relations; derived relations; quotient structure ==

Revision as of 23:10, 18 May 2008

A quotient set is a set derived from another by an equivalence relation.

Let $S$ be a set, and let $\mathcal{R}$ be an equivalence relation. The set of equivalence classes of $S$ with respect to $\mathcal{R}$ is called the quotient of $S$ by $\mathcal{R}$, and is denoted $S/\mathcal{R}$.

A subset $A$ of $S$ is said to be saturated with respect to $\mathcal{R}$ if for all $x,y \in S$, $x\in A$ and $\mathcal{R}(x,y)$ imply $y\in A$. Equivalently, $A$ is saturated if it is the union of a family of equivalence classes with respect to $\mathcal{R}$. The saturation of $A$ with respect to $\mathcal{R}$ is the least saturated subset $A'$ of $S$ that contains $A$.

Compatible relations; derived relations; quotient structure

Let $P(x)$ be a relation, and let $\mathcal{R}$ be an equivalence relation. If $\mathcal{R}(x,y)$ and $P(x)$ together imply $P(y)$, then $P$ is said to be compatible with $\mathcal{R}$.

Let $P(x)$ be a relation. The relation $P'(y)$ on the elements of $S/\mathcal{R}$, defined as 

\[\exist x\in y, P(x)\] (Error compiling LaTeX. Unknown error_msg)

is called the relation derived from $P$ by passing to the quotient.

Let $S$ be a structure, $\mathcal{R}$, an equivalence relation. If the equivalence classes form a structure of the same species as $S$ under relations derived from passing to quotients, $\mathcal{R}$ is said to be compatible with the structure on $S$, and this structure on the equivalence classes of $S$ is called the quotient structure, or the derived structure, of $S/\mathcal{R}$.

This article is a stub. Help us out by expanding it.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Quotient_set&oldid=26043"