Notes on quotient sets, function kernels, and fibers.

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@@ -143,7 +143,8 @@
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notes/_journal/2024-07-17.md

@@ -0,0 +1,11 @@
+---
+title: "2024-07-17"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[relations#Quotient Sets|quotient sets]], function [[functions#Kernels|kernels]], and fibers.

notes/_journal/2024-07/2024-07-14.md

@@ -11,4 +11,5 @@ title: "2024-07-14"
 * Notes on [[set#Cartesian Product|infinite Cartesian products]] and their relation to the [[set/index#Infinite Cartesian Product Form|axiom of choice]].
 * Initial notes on [[relations#Equivalence Relations|equivalence relations]].
 * Read chapter 2 "How to Raise Money" in "Venture Deals".
-* Finished another read of "A Cardinal Worry for Permissive Metaontology".
+* Finished another read of "A Cardinal Worry for Permissive Metaontology".
+* Watched [Lecture 1 "Introduction to Ontology"](https://www.youtube.com/watch?v=9AsRE437e7I).

notes/_journal/2024-07/2024-07-15.md

@@ -0,0 +1,11 @@
+---
+title: "2024-07-15"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[relations#Partitions|partitions]] and equivalence classes.

notes/_journal/2024-07/2024-07-16.md

@@ -0,0 +1,11 @@
+---
+title: "2024-07-16"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Brief notes on [[nominalism]].

notes/algebra/set.md

@@ -93,7 +93,7 @@ END%%
 
 %%ANKI
 Basic
-Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
+Suppose $x, y \in A$. What set, derived from $A$, is $\langle x, y \rangle$ a member of?
 Back: $\mathscr{P}\mathscr{P}A$
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1717679397848-->
@@ -758,8 +758,8 @@ Let $A$, $B$, and $C$ be arbitrary sets. Then
 
 %%ANKI
 Basic
-What kind of propositional logical statement are the monotonicity properties of $\subseteq$?
+The monotonicity properties of $\subseteq$ are what kind of propositional logical statement?
-Back: An implication.
+Back: Implications.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1717073536967-->
 END%%

notes/algorithms/order-growth.md

@@ -690,7 +690,7 @@ END%%
 %%ANKI
 Basic
 What names are usually given to the existentially quantified identifers in $o(g(n))$'s definition?
-Back: $n_0$.
+Back: $n_0$
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1709519002328-->
 END%%
@@ -1035,7 +1035,7 @@ END%%
 %%ANKI
 Basic
 What is the symmetric property of $\Omega$-notation?
-Back: N/A
+Back: N/A.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1709752223486-->
 END%%

notes/encoding/integer.md

@@ -557,7 +557,7 @@ END%%
 
 %%ANKI
 Basic
-What is the precise definition of the two's-complement of a $w$-bit number?
+What is the precise definition of the two's-complement of a $w$-bit number $x$?
 Back: The complement of $x$ with respect to $2^w$.
 Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
 <!--ID: 1709060837145-->

notes/hashing/index.md

@@ -404,7 +404,7 @@ END%%
 %%ANKI
 Basic
 Let $h$ be a division method hash function. What does $h(10)$ evaluate to?
-Back: $10 \bmod{m}$ where $m$ is the number of slots in the hash table.
+Back: To $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 Tags: hashing::static
 <!--ID: 1720889385419-->
@@ -428,6 +428,14 @@ Tags: hashing::static
 <!--ID: 1720889385429-->
 END%%
 
+%%ANKI
+Basic
+Why does the division method prefer a prime number of slots?
+Back: To operate as independently as possible of the input keys.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1721218408542-->
+END%%
+
 %%ANKI
 Basic
 Consider hash function $h(k) = k \bmod{m}$. What method was likely used to produce this?

notes/lambda-calculus/beta-reduction.md

@@ -567,7 +567,7 @@ END%%
 
 %%ANKI
 Basic
-What does the Church-Rosser theorem state in terms of confluence?
+What does the Church-Rosser theorem for $\triangleright_\beta$ state in terms of confluence?
 Back: $\beta$-reduction is confluent.
 Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
 <!--ID: 1719577152613-->

notes/logic/truth-tables.md

@@ -50,7 +50,7 @@ $$
 %%ANKI
 Basic
 What construct is used to prove every proposition can be written in DNF or CNF?
-Back: Truth tables
+Back: Truth tables.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1707311868994-->
 END%%

notes/ontology/index.md

@@ -21,7 +21,7 @@ END%%
 %%ANKI
 Basic
 Who is attributed *the* ontological question?
-Back: Quine.
+Back: Willard Van Orman Quine.
 Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009).
 <!--ID: 1720912259767-->
 END%%

notes/ontology/nominalism.md

@@ -0,0 +1,47 @@
+---
+title: Nominalism
+TARGET DECK: Obsidian::H&SS
+FILE TAGS: ontology::nominalism
+tags:
+  - nominalism
+  - ontology
+---
+
+## Overview
+
+**Anti-realists** about a category are those who don't believe entities of said category exist. **Realists** about a category are those that do. **Nominalism** refers to the stance that no abstract objects exist. That is, nominalists are anti-realists in regards to abstract entities.
+
+%%ANKI
+Basic
+What does it mean for a person to be anti-realist about category $X$?
+Back: They do not believe entities of $X$ exist.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1721137271035-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a person to be realist about category $X$?
+Back: They believe entities belonging to $X$ exist.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1721137271065-->
+END%%
+
+%%ANKI
+Basic
+How does Effingham define nominalism?
+Back: As anti-realism towards abstracta.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1721137271072-->
+END%%
+
+%%ANKI
+Cloze
+Roughly speaking, {1:permissivism} is to {2:realism} whereas {2:nominalism} is to {1:anti-realism}.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1721137271080-->
+END%%
+
+## Bibliography
+
+* Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).

notes/set/functions.md

@@ -1327,7 +1327,7 @@ END%%
 
 %%ANKI
 Basic
-Consider sets $A$ and $B$. How is $A \cap B$ be rewritten as a function under some image?
+Consider sets $A$ and $B$. How is $A \cap B$ rewritten as a function under some image?
 Back: $I_A[\![B]\!]$
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1720885546358-->
@@ -1634,7 +1634,174 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1720819771087-->
 END%%
 
+## Kernels
+
+Let $F \colon A \rightarrow B$. Define [[relations#Equivalence Relations|equivalence relation]] $\sim$ as $$x \sim y \Leftrightarrow f(x) = f(y)$$
+Relation $\sim$ is called the **(equivalence) kernel** of $f$. The [[relations#Partitions|partition]] induced by $\sim$ on $A$ is called the **coimage** of $f$ (denoted $\mathop{\text{coim}}f$). The **fiber** of an element $y$ under $F$ is $F^{-1}[\![\{y\}]\!]$, i.e. the preimage of singleton set $\{y\}$. Therefore the equivalence classes of $\sim$ are also known as the fibers of $f$.
+
+%%ANKI
+Basic
+What kind of mathematical object is the kernel of $F \colon A \rightarrow B$?
+Back: An equivalence relation.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015583-->
+END%%
+
+%%ANKI
+Basic
+How is the kernel of $F \colon A \rightarrow B$ defined?
+Back: As equivalence relation $\sim$ such that $x \sim y \Leftrightarrow F(x) = F(y)$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015586-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$. What name does the following relation $\sim$ go by? $$x \sim y \Leftrightarrow F(x) = F(y)$$
+Back: The kernel of $F$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015590-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$. The partition induced by the kernel of $F$ is a partition of what set?
+Back: $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015593-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$. What does $\mathop{\text{coim}}F$ refer to?
+Back: The coimage of $F$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015596-->
+END%%
+
+%%ANKI
+Basic
+How is the coimage of function $F \colon A \rightarrow B$ defined?
+Back: As $A / {\sim}$ where $x \sim y \Leftrightarrow F(x) = F(y)$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015599-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$. What specific name does a member of $\mathop{\text{coim}}F$ go by?
+Back: A fiber.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015602-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$. How is the fiber of $y$ under $F$ defined?
+Back: As set $F^{-1}[\![\{y\}]\!]$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015605-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$. The fibers of $F$ make up what set?
+Back: $\mathop{\text{coim}}F$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015609-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$. How is $\mathop{\text{coim}}F$ denoted as a quotient set?
+Back: As $A / {\sim}$ where $x \sim y \Leftrightarrow F(x) = F(y)$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015613-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$ and $\sim$ be the kernel of $F$. How does $F$ factor into $\hat{F} \colon A / {\sim} \rightarrow B$?
+Back: $F = \hat{F} \circ \phi$ where $\phi$ is the natural map.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015617-->
+END%%
+
+%%ANKI
+Basic
+Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What name does $\phi$ go by?
+![[function-kernel.png]]
+Back: The natural map (with respect to $\sim$).
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015620-->
+END%%
+
+%%ANKI
+Basic
+Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. How is $\phi$ defined?
+![[function-kernel.png]]
+Back: $\phi(x) = [x]_{\sim}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015624-->
+END%%
+
+%%ANKI
+Basic
+Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What name does $\sim$ go by?
+![[function-kernel.png]]
+Back: $\mathop{\text{coim}} F$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015628-->
+END%%
+
+%%ANKI
+Basic
+Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What name do the members of $A / {\sim}$ go by?
+![[function-kernel.png]]
+Back: The fibers of $F$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015633-->
+END%%
+
+%%ANKI
+Basic
+Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. What composition is $F$ equal to?
+![[function-kernel.png]]
+Back: $F = \hat{F} \circ \phi$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015638-->
+END%%
+
+%%ANKI
+Basic
+Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. Is $\hat{F}$ injective?
+![[function-kernel.png]]
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015642-->
+END%%
+
+%%ANKI
+Basic
+Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. Is $\hat{F}$ surjective?
+![[function-kernel.png]]
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015646-->
+END%%
+
+%%ANKI
+Basic
+Consider factoring $F \colon A \rightarrow B$ by its kernel $\sim$. Is $\hat{F}$ bijective?
+![[function-kernel.png]]
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015650-->
+END%%
+
 ## Bibliography
 
 * “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
-* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+* “Fiber (Mathematics),” in _Wikipedia_, April 10, 2024, [https://en.wikipedia.org/w/index.php?title=Fiber_(mathematics)&oldid=1218193490](https://en.wikipedia.org/w/index.php?title=Fiber_(mathematics)&oldid=1218193490).
+* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+* “Kernel (Set Theory),” in _Wikipedia_, May 22, 2024, [https://en.wikipedia.org/w/index.php?title=Kernel_(set_theory)&oldid=1225189560](https://en.wikipedia.org/w/index.php?title=Kernel_(set_theory)&oldid=1225189560).

notes/set/relations.md

@@ -803,6 +803,331 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1720969371869-->
 END%%
 
+The set $[x]_R$ is defined by $[x]_R = \{t \mid xRt\}$. If $R$ is an equivalence relation and $x \in \mathop{\text{fld}}R$, then $[x]_R$ is called the **equivalence class of $x$ (modulo $R$)**. If the relation $R$ is fixed by the context, we may write just $[x]$.
+
+%%ANKI
+Basic
+How is set $[x]_R$ defined?
+Back: As $\{t \mid xRt\}$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094107-->
+END%%
+
+%%ANKI
+Basic
+What is an equivalence class?
+Back: A set of members mutually related w.r.t an equivalence relation.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015574-->
+END%%
+
+%%ANKI
+Basic
+What kind of mathematical object is $x$ in $[x]_R$?
+Back: A set (or urelement).
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094110-->
+END%%
+
+%%ANKI
+Basic
+What kind of mathematical object is $R$ in $[x]_R$?
+Back: A relation.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094114-->
+END%%
+
+%%ANKI
+Basic
+What compact notation is used to denote $\{t \mid xRt\}$?
+Back: $[x]_R$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094120-->
+END%%
+
+%%ANKI
+Cloze
+If {1:$R$ is an equivalence relation} and {1:$x \in \mathop{\text{fld} }R$}, then $[x]_R$ is called the {2:equivalence class of $x$} (modulo {2:$R$}).
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094128-->
+END%%
+
+%%ANKI
+Basic
+Consider an equivalence class of $x$ (modulo $R$). What kind of mathematical object is $x$?
+Back: A set (or urelement).
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094137-->
+END%%
+
+%%ANKI
+Basic
+Consider an equivalence class of $x$ (modulo $R$). What kind of mathematical object is $R$?
+Back: A relation.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094144-->
+END%%
+
+%%ANKI
+Basic
+Consider an equivalence class of $x$ (modulo $R$). What condition does $x$ necessarily satisfy?
+Back: $x \in \mathop{\text{fld}}R$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094149-->
+END%%
+
+%%ANKI
+Basic
+Consider an equivalence class of $x$ (modulo $R$). What condition does $R$ necessarily satisfy?
+Back: $R$ is an equivalence relation.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094154-->
+END%%
+
+%%ANKI
+Cloze
+Assume $R$ is an equivalence relation on $A$ and that $x, y \in A$. Then {1:$[x]_R$} $=$ {1:$[y]_R$} iff {2:$xRy$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094158-->
+END%%
+
+## Partitions
+
+A **partition** $\Pi$ of a set $A$ is a set of nonempty subsets of $A$ that is disjoint and exhaustive.
+
+%%ANKI
+Basic
+What kind of mathematical object is a partition of a set?
+Back: A set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094026-->
+END%%
+
+%%ANKI
+Basic
+What is a partition of a set $A$?
+Back: A set of nonempty subsets of $A$ that is disjoint and exhaustive.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094053-->
+END%%
+
+%%ANKI
+Basic
+Let $\Pi$ be a partition of a set $A$. When does $\Pi = \varnothing$?
+Back: If and only if $A = \varnothing$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094059-->
+END%%
+
+%%ANKI
+Basic
+Let $\Pi$ be a partition of set $A$. What property must each *individual* member of $\Pi$ exhibit?
+Back: Each member is nonempty.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094065-->
+END%%
+
+%%ANKI
+Basic
+Let $\Pi$ be a partition of set $A$. What property must each *pair* of members of $\Pi$ exhibit?
+Back: Each pair must be disjoint.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094072-->
+END%%
+
+%%ANKI
+Basic
+Let $\Pi$ be a partition of set $A$. Which property do all the members of $\Pi$ exhibit together?
+Back: The members of $\Pi$ must be exhaustive.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094077-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a partition $\Pi$ of $A$ to be exhaustive?
+Back: Every member of $A$ must appear in one of the members of $\Pi$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094082-->
+END%%
+
+%%ANKI
+Basic
+Is $A$ a partition of set $A$?
+Back: No.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094086-->
+END%%
+
+%%ANKI
+Basic
+Is $\{A\}$ a partition of set $A$?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094091-->
+END%%
+
+%%ANKI
+Basic
+Let $A = \{1, 2, 3, 4\}$. Why isn't $\{\{1, 2\}, \{2, 3, 4\}\}$ a partition of $A$?
+Back: Each pair of members of a partition of $A$ must be disjoint.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094095-->
+END%%
+
+%%ANKI
+Basic
+Let $A = \{1, 2, 3, 4\}$. Why isn't $\{\{1\}, \{2\}, \{3\}\}$ a partition of $A$?
+Back: The members of a partition of $A$ must be exhaustive.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094099-->
+END%%
+
+%%ANKI
+Basic
+Let $A = \{1, 2, 3, 4\}$. Why isn't $\{\{1, 2, 3\}, \{4\}\}$ a partition of $A$?
+Back: N/A. It is.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721098094103-->
+END%%
+
+Assume $\Pi$ is a partition of set $A$. Then the relation $R$ is an equivalence relation: $$xRy \Leftrightarrow (\exists B \in \Pi, x \in B \land y \in B)$$
+
+%%ANKI
+Basic
+Let $\Pi$ be a partition of $A$. What equivalence relation $R$ is induced?
+Back: $R$ such that $xRy \Leftrightarrow (\exists B \in \Pi, x \in B \land y \in B)$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721136390215-->
+END%%
+
+## Quotient Sets
+
+If $R$ is an equivalence relation on $A$, then the **quotient set** "$A$ modulo $R$" is defined as $$A / R = \{[x]_R \mid x \in A\}.$$
+
+The **natural map** (or **canonical map**) $\phi : A \rightarrow A / R$ is given by $$\phi(x) = [x]_R.$$
+
+Note that $A / R$, the set of all equivalence classes, is a partition of $A$.
+
+%%ANKI
+Basic
+Let $R$ be an equivalence relation on $A$. What partition is induced?
+Back: $A / R = \{[x]_R \mid x \in A\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721136390208-->
+END%%
+
+%%ANKI
+Basic
+Members of $A / R$ are called what?
+Back: Equivalence classes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408454-->
+END%%
+
+%%ANKI
+Basic
+$A / R$ is a partition of what set?
+Back: $A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408484-->
+END%%
+
+%%ANKI
+Basic
+How is quotient set $A / R$ pronounced?
+Back: As "$A$ modulo $R$".
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408508-->
+END%%
+
+%%ANKI
+Basic
+Consider quotient set $A / R$. What kind of mathematical object is $A$?
+Back: A set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408514-->
+END%%
+
+%%ANKI
+Basic
+Consider quotient set $A / R$. What kind of mathematical object is $R$?
+Back: An equivalence relation on $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408520-->
+END%%
+
+%%ANKI
+Basic
+How is quotient set $A / R$ defined?
+Back: As set $\{[x]_R \mid x \in A\}$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408525-->
+END%%
+
+%%ANKI
+Basic
+Given quotient set $A / R$, what is the domain of its natural map?
+Back: $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408490-->
+END%%
+
+%%ANKI
+Basic
+Given quotient set $A / R$, what is the codomain of its natural map?
+Back: $A / R$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408495-->
+END%%
+
+%%ANKI
+Basic
+Consider quotient set $A / R$. How is the natural map $\phi$ defined?
+Back: $\phi \colon A \rightarrow A / R$ given by $\phi(x) = [x]_R$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408501-->
+END%%
+
+%%ANKI
+Basic
+Given quotient set $A / R$, what is the domain of its canonical map?
+Back: $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408531-->
+END%%
+
+%%ANKI
+Basic
+Given quotient set $A / R$, what is the codomain of its canonical map?
+Back: $A / R$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218408537-->
+END%%
+
+%%ANKI
+Basic
+Consider quotient set $A / R$. How is the canonical map $\phi$ defined?
+Back: $\phi \colon A \rightarrow A / R$ given by $\phi(x) = [x]_R$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721218465987-->
+END%%
+
+%%ANKI
+Basic
+Consider set $\omega$ and equivalence relation $\sim$. How is the relevant quotient set denoted?
+Back: As $\omega / {\sim}$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721219061765-->
+END%%
+
+%%ANKI
+Cloze
+Let $R$ be an equivalence relation on $A$ and $x \in A$. Then {1:$x$ (modulo $R$)} is an {2:equivalence class} whereas {2:$A$ modulo $R$} is a {1:quotient set}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1721223015580-->
+END%%
+
 ## Bibliography
 
 * “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).