Archimedean property and natural numbers.

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notes/_journal/2024-08-24.md

@@ -0,0 +1,11 @@
+---
+title: "2024-08-24"
+---
+
+- [ ] Anki Flashcards
+- [ ] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Began notes on [[natural-numbers|natural numbers]].

notes/_journal/2024-08/2024-08-20.md

@@ -0,0 +1,11 @@
+---
+title: "2024-08-20"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Add flashcards on the [[arch-prop|Archimedean property]] of reals.

notes/_journal/2024-08/2024-08-21.md

@@ -0,0 +1,9 @@
+---
+title: "2024-08-21"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/_journal/2024-08/2024-08-22.md

@@ -0,0 +1,11 @@
+---
+title: "2024-08-22"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Watched [Lecture 11 - Join Algorithms](https://www.youtube.com/watch?v=yFk_GfaY2Hk&list=PLSE8ODhjZXjaKScG3l0nuOiDTTqpfnWFf).

notes/_journal/2024-08/2024-08-23.md

@@ -0,0 +1,9 @@
+---
+title: "2024-08-23"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/algebra/arch-prop.md

@@ -0,0 +1,98 @@
+---
+title: Archimedean Property
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algebra::archimedean
+tags:
+  - algebra
+---
+
+## Overview
+
+If $x, y \in \mathbb{R}^+$, then there exists a positive integer $n$ such that $nx > y$. This fundamental property usually follows from the [[bounds#Completeness Axiom|completeness axiom]].
+
+%%ANKI
+Basic
+What does the Archimedean property of the reals state?
+Back: If $x, y \in \mathbb{R}^+$, then there exists a positive integer $n$ such that $nx > y$.
+Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
+<!--ID: 1724155425457-->
+END%%
+
+%%ANKI
+Basic
+How is the Archimedean property of the reals geometrically interpreted?
+Back: Any finite-length line segment can be covered by a finite number of line segments of some positive length.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724155425465-->
+END%%
+
+%%ANKI
+Basic
+The Archimedean property of the reals posits the existence of what mathematical entity?
+Back: A positive integer.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724155425470-->
+END%%
+
+%%ANKI
+Basic
+Given positive reals $x$ and $y$, what does the Archimedean property conclude?
+Back: There exists a positive integer $n$ such that $nx > y$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724155425481-->
+END%%
+
+%%ANKI
+Basic
+Given reals $x$ and $y$, what does the Archimedean property conclude?
+Back: Indeterminate. We expect $x$ and $y$ to be positive reals.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724155425487-->
+END%%
+
+%%ANKI
+Basic
+Which property is roughly described as "the reals have no infinitely large element?"
+Back: The Archimedean property of the reals.
+Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
+<!--ID: 1724155425492-->
+END%%
+
+%%ANKI
+Basic
+Which property of the reals is depicted in the following?
+![[archimedean-property.png]]
+Back: The Archimedean property.
+Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
+<!--ID: 1724155425498-->
+END%%
+
+%%ANKI
+Basic
+How is the following diagram stated analytically?
+![[archimedean-property.png]]
+Back: For any $A, B \in \mathbb{R}^+$, there exists a positive integer $n$ such that $nA > B$.
+Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
+<!--ID: 1724155425503-->
+END%%
+
+%%ANKI
+Basic
+What mathematical entities are assumed to exist in the formulate of the Archimedean property of the reals?
+Back: Two positive real numbers.
+Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
+<!--ID: 1724155425509-->
+END%%
+
+%%ANKI
+Basic
+What axiom of the real-number system is used to prove its Archimedean property?
+Back: The least upper bound axiom (i.e. the completeness axiom).
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1724155536942-->
+END%%
+
+## Bibliography
+
+* “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
+* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).

notes/algebra/set.md

@@ -1081,14 +1081,6 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1720782833233-->
 END%%
 
-%%ANKI
-Basic
-For any function $F \colon A \rightarrow B$, $F$ is a member of what other set?
-Back: $\mathscr{P}(A \times B)$
-Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1720782833236-->
-END%%
-
 %%ANKI
 Basic
 For sets $A$ and $B$, how is set $B^A$ pronounced?

notes/c17/index.md

@@ -15,7 +15,7 @@ This quote describes C's **abstract state machine**. Whatever instructions a C p
 %%ANKI
 Basic
 What feature of C's abstract state machine makes C performant?
-Back: It enables optimization.
+Back: The ability to optimize.
 Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
 <!--ID: 1723856661330-->
 END%%

notes/c17/types.md

@@ -27,7 +27,7 @@ END%%
 %%ANKI
 Basic
 Why are narrow types named the way they are?
-Back: They are considered to small to be used directly in arithmetic expressions.
+Back: They are considered too small to be used directly in arithmetic expressions.
 Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
 <!--ID: 1723859121959-->
 END%%

notes/programming/pred-trans.md

@@ -979,7 +979,7 @@ END%%
 %%ANKI
 Basic
 When are both of the following guarded commands executed? $$\begin{align*} \textbf{if } & x \geq 0 \rightarrow z \coloneqq x \\ \textbf{ | } & x \leq 0 \rightarrow z \coloneqq -x \\ \textbf{fi } & \end{align*}$$
-Back: N/A.
+Back: N/A. Only one guard is executed.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1722257348960-->
 END%%

notes/set/index.md

@@ -1115,6 +1115,33 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1720964209648-->
 END%%
 
+## Infinity Axiom
+
+There exists an [[natural-numbers#Inductive Sets|inductive]] set: $$\exists A, [\varnothing \in A \land (\forall a \in A, a^+ \in A)]$$
+
+%%ANKI
+Basic
+What does the infinity axiom state?
+Back: There exists an inductive set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269578-->
+END%%
+
+%%ANKI
+Cloze
+The {infinity axiom} asserts the existence of an {inductive set}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269581-->
+END%%
+
+%%ANKI
+Basic
+State the infinity axiom in FOL.
+Back: $\exists A, [\varnothing \in A \land (\forall a \in A, a^+ \in A)]$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269585-->
+END%%
+
 ## Bibliography
 
 * “Axiom of Choice,” in _Wikipedia_, July 8, 2024, [https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262](https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262).

notes/set/natural-numbers.md

@@ -0,0 +1,286 @@
+---
+title: Natural Numbers
+TARGET DECK: Obsidian::STEM
+FILE TAGS: set::nat
+tags:
+  - natural-number
+  - set
+---
+
+## Overview
+
+The standard way of representing the natural numbers is as follows:
+
+* $0 = \varnothing$
+* $1 = \{0\} = \{\varnothing\}$
+* $2 = \{0, 1\} = \{\varnothing, \{\varnothing\}\}$
+* $\ldots$
+
+That is, each natural number corresponds to the set of natural numbers smaller than it.
+
+%%ANKI
+Basic
+How is the number $0$ represented as a set?
+Back: $\varnothing$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233219-->
+END%%
+
+%%ANKI
+Basic
+How is the number $1$ represented as a set?
+Back: $\{0\} = \{\varnothing\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233247-->
+END%%
+
+%%ANKI
+Basic
+How is the number $2$ represented as a set?
+Back: $\{0, 1\} = \{\varnothing, \{\varnothing\}\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233252-->
+END%%
+
+%%ANKI
+Basic
+Who came up with the standard set representation of natural numbers?
+Back: John von Neumann.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233257-->
+END%%
+
+%%ANKI
+Basic
+Consider the set representation of $n \in \mathbb{N}$. How many members does $n$ have?
+Back: $n$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233263-->
+END%%
+
+%%ANKI
+Basic
+Consider the set representation of $n \in \mathbb{N}$. What are the members of $n$?
+Back: $0$, $1$, $\ldots$, $n - 1$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233269-->
+END%%
+
+%%ANKI
+Basic
+Let $n \in \mathbb{N}$. *Why* is $n \in n + 1$?
+Back: $n + 1$ is a set containing all preceding natural numbers.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233274-->
+END%%
+
+%%ANKI
+Basic
+Let $n \in \mathbb{N}$. *Why* is $n \subseteq n + 1$?
+Back: $n$ and $n + 1$ are sets containing all their preceding natural numbers.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233279-->
+END%%
+
+## Inductive Sets
+
+For any set $a$, its **successor** $a^+$ is defined as $$a^+ = a \cup \{a\}$$
+
+%%ANKI
+Basic
+How is the successor of a set $a$ denoted?
+Back: $a^+$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233287-->
+END%%
+
+%%ANKI
+Basic
+How is the successor of a set $a$ defined?
+Back: As $a^+ = a \cup \{a\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233291-->
+END%%
+
+%%ANKI
+Basic
+Set $\{a, b\}^+$ equals what other set?
+Back: $\{a, b, \{a, b\}\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233295-->
+END%%
+
+%%ANKI
+Basic
+Set $\{a\}^+$ equals what other set?
+Back: $\{a, \{a\}\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233299-->
+END%%
+
+%%ANKI
+Basic
+Set $\{a, \{a, b\}, \{a, b, c\}\}$ can be written as the successor of what set?
+Back: N/A.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485516768-->
+END%%
+
+%%ANKI
+Basic
+Set $\{a, b, \{a, b\}\}$ can be written as the successor of what set?
+Back: $\{a, b\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485516774-->
+END%%
+
+%%ANKI
+Basic
+Set $\{a, \{a, b\}\}$ can be written as the successor of what set?
+Back: N/A.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485516777-->
+END%%
+
+%%ANKI
+Basic
+Set $\{a, \{a, b\}, \{a, \{a, b\}\}\}$ can be written as the successor of what set?
+Back: $\{a, \{a, b\}\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485516780-->
+END%%
+
+%%ANKI
+Basic
+If $n \in \mathbb{N}$ then $n \in n + 1$. What analagous statement holds for arbitrary set $a$?
+Back: $a \in a^+$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233303-->
+END%%
+
+%%ANKI
+Basic
+If $n \in \mathbb{N}$ then $n \subseteq n + 1$. What analagous statement holds for arbitrary set $a$?
+Back: $a \subseteq a^+$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724485233283-->
+END%%
+
+A set $A$ is **inductive** if and only if $\varnothing \in A$ and $\forall a \in A, a^+ \in A$.
+
+%%ANKI
+Basic
+What does it mean for a set $A$ to be closed under successor?
+Back: If $a \in A$, then $a^+ \in A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269548-->
+END%%
+
+%%ANKI
+Basic
+Write "set $B$ is closed under successor" in FOL.
+Back: $\forall b \in B, b^+ \in B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269552-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a set $A$ to be inductive?
+Back: $\varnothing \in A$ and $A$ is closed under successor.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269555-->
+END%%
+
+%%ANKI
+Cloze
+A set $A$ is inductive iff {$\varnothing \in A$} and {$A$ is closed under successor}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269558-->
+END%%
+
+%%ANKI
+Basic
+An inductive set is closed under what operation?
+Back: Successor.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269562-->
+END%%
+
+%%ANKI
+Basic
+What set is the "seed" of an inductive set?
+Back: $\varnothing$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269565-->
+END%%
+
+%%ANKI
+Basic
+Let $a \in A$ where $A$ is an inductive set. What other members must belong to $A$?
+Back: $a^+$, $a^{++}$, $\ldots$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269568-->
+END%%
+
+%%ANKI
+Basic
+What natural number corresponds to $\varnothing^{+++}$?
+Back: $3$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269571-->
+END%%
+
+%%ANKI
+Basic
+What natural number corresponds to $\varnothing$?
+Back: $0$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486269575-->
+END%%
+
+A **natural number** is a set that belongs to every inductive set.
+
+%%ANKI
+Basic
+How is a natural number *defined* in set theory?
+Back: As a set belonging to every inductive set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486756997-->
+END%%
+
+%%ANKI
+Basic
+What greek letter is used to denote the set of natural numbers?
+Back: $\omega$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486757001-->
+END%%
+
+%%ANKI
+Basic
+What is the smallest inductive set?
+Back: $\omega$, i.e. the set of natural numbers.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486757004-->
+END%%
+
+%%ANKI
+Basic
+How might $\omega$ be defined as an intersection of classes?
+Back: $\omega = \bigcap\,\{A \mid A \text{ is inductive}\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486757007-->
+END%%
+
+%%ANKI
+Basic
+Suppose $n \in \omega$. What other sets *must* $n$ be a member of?
+Back: Every other inductive set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1724486757010-->
+END%%
+
+## Bibliography
+
+* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

notes/set/order.md

@@ -89,7 +89,7 @@ END%%
 
 %%ANKI
 Cloze
-Operator {$\leq$} typically denote a {non-strict} preorder.
+Operator {$\leq$} typically denotes a {non-strict} preorder.
 Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
 <!--ID: 1723924394146-->
 END%%
@@ -177,7 +177,7 @@ END%%
 
 %%ANKI
 Cloze
-Operator {$<$} typically denote a {strict} preorder.
+Operator {$<$} typically denotes a {strict} preorder.
 Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
 <!--ID: 1723924394151-->
 END%%
@@ -268,7 +268,7 @@ END%%
 %%ANKI
 Basic
 *Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ a partial order on $\{a, b, c\}$?
-Back: N/A. It is.
+Back: It isn't reflexive on $\{b, c\}$.
 Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
 <!--ID: 1723816108524-->
 END%%
@@ -276,7 +276,7 @@ END%%
 %%ANKI
 Basic
 *Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle, \langle c, b \rangle\}$ a partial order on $\{a, b, c\}$?
-Back: It isn't antisymmetric.
+Back: It isn't reflexive on $\{b, c\}$, it isn't antisymmetric, and it isn't transitive.
 Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
 <!--ID: 1723816108531-->
 END%%
@@ -309,14 +309,14 @@ END%%
 
 %%ANKI
 Cloze
-Operator {$<$} typically denote a {strict} partial order.
+Operator {$<$} typically denotes a {strict} partial order.
 Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
 <!--ID: 1723902024378-->
 END%%
 
 %%ANKI
 Cloze
-Operator {$\leq$} typically denote a {non-strict} partial order.
+Operator {$\leq$} typically denotes a {non-strict} partial order.
 Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
 <!--ID: 1723902024382-->
 END%%
@@ -414,7 +414,7 @@ END%%
 %%ANKI
 Basic
 *Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ an equivalence relation on $\{a, b\}$?
-Back: It isn't symmetric.
+Back: It is neither reflexive on $\{a, b\}$ nor symmetric.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1723816108538-->
 END%%
@@ -855,7 +855,7 @@ END%%
 
 %%ANKI
 Cloze
-Operator {$\leq$} typically denote a {non-strict} total order.
+Operator {$\leq$} typically denotes a {non-strict} total order.
 Reference: “Total Order.” In _Wikipedia_, April 9, 2024. [https://en.wikipedia.org/w/index.php?title=Total_order](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468).
 <!--ID: 1723923665325-->
 END%%
@@ -896,7 +896,7 @@ END%%
 
 %%ANKI
 Cloze
-Operator {$<$} typically denote a {strict} total order.
+Operator {$<$} typically denotes a {strict} total order.
 Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
 <!--ID: 1723923665330-->
 END%%
@@ -918,7 +918,7 @@ END%%
 
 %%ANKI
 Cloze
-A {non-strict} total order satisfies {strong connectivity} whereas a {strict} total order satisfies {connectivity}.
+A {1:non-strict} total order satisfies {2:strong connectivity} whereas a {2:strict} total order satisfies {1:connectivity}.
 Reference: “Total Order.” In _Wikipedia_, April 9, 2024. [https://en.wikipedia.org/w/index.php?title=Total_order](https://en.wikipedia.org/w/index.php?title=Total_order&oldid=1218090468).
 <!--ID: 1723923665333-->
 END%%

notes/x86-64/registers.md

@@ -238,35 +238,35 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 END%%
 
 %%ANKI
-Cloze
+Basic
 Which register should I use for an 2 byte return value?
 Back: `%ax`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1724119420103-->
+<!--ID: 1724417749880-->
 END%%
 
 %%ANKI
-Cloze
+Basic
 Which register should I use for a 1 byte stack pointer?
 Back: `%spl`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1724119420106-->
+<!--ID: 1724417749885-->
 END%%
 
 %%ANKI
-Cloze
+Basic
 Which register should I use for a 4 byte stack pointer?
 Back: `%esp`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1724119420109-->
+<!--ID: 1724417749890-->
 END%%
 
 %%ANKI
-Cloze
+Basic
 Which register should I use for an 8 byte return value?
 Back: `%rax`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1724119420113-->
+<!--ID: 1724417749896-->
 END%%
 
 %%ANKI