Notes on ordered pairs.

2024-06-06 07:14:13 -06:00 · 2024-06-06 07:14:13 -06:00 · cfa95d2390
parent 9c39cc2b49
commit cfa95d2390
11 changed files with 359 additions and 20 deletions
--- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
+++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
@ -133,7 +133,8 @@
    "venn-diagram-union.png",
    "venn-diagram-intersection.png",
    "venn-diagram-rel-comp.png",
-    "venn-diagram-abs-comp.png"
+    "venn-diagram-abs-comp.png",
+    "venn-diagram-symm-diff.png"
  ],
  "File Hashes": {
    "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8",
@ -313,7 +314,7 @@
    "_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f",
    "_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b",
    "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53",
-    "set/index.md": "55a80d53973af2c0343438346e2086b2",
+    "set/index.md": "d1fcdfe4506aab4ec86f30118464fb2a",
    "set/graphs.md": "4bbcea8f5711b1ae26ed0026a4a69800",
    "_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb",
    "_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391",
@ -332,7 +333,7 @@
    "_journal/2024-03-22.md": "8da8cda07d3de74f7130981a05dce254",
    "_journal/2024-03/2024-03-21.md": "cd465f71800b080afa5c6bdc75bf9cd3",
    "x86-64/declarations.md": "75bc7857cf2207a40cd7f0ee056af2f2",
-    "x86-64/instructions.md": "c9ae5dedaa22fbcdc486663877fc1c1e",
+    "x86-64/instructions.md": "06b7fbe1a7a9568b80239310eb72e87a",
    "git/refs.md": "e20c2c9b14ba6c2bd235416017c5c474",
    "set/trees.md": "b085bd8d08dccd8cbf02bb1b7a4baa43",
    "_journal/2024-03-24.md": "1974cdb9fc42c3a8bebb8ac76d4b1fd6",
@ -466,7 +467,7 @@
    "programming/λ-Calculus.md": "bf36bdaf85abffd171bb2087fb8228b2",
    "_journal/2024-05-23.md": "9d9106a68197adcee42cd19c69d2f840",
    "_journal/2024-05/2024-05-22.md": "3c29eec25f640183b0be365e7a023750",
-    "programming/lambda-calculus.md": "547bc98111e516fc70a415d91db7b3ff",
+    "programming/lambda-calculus.md": "89f024561ea770552bc20056a45dbce6",
    "_journal/2024-05-25.md": "04e8e1cf4bfdbfb286effed40b09c900",
    "_journal/2024-05/2024-05-24.md": "86132f18c7a27ebc7a3e4a07f4867858",
    "_journal/2024-05/2024-05-23.md": "d0c98b484b1def3a9fd7262dcf2050ad",
@ -474,8 +475,8 @@
    "_journal/2024-05/2024-05-25.md": "3e8a0061fa58a6e5c48d12800d1ab869",
    "_journal/2024-05-27.md": "b36636d10eab34380f17f288868df3ae",
    "_journal/2024-05/2024-05-26.md": "abe84b5beae74baa25501c818e64fc95",
-    "algebra/set.md": "049c5ff36ce8fe99b2c8a8f452d0514f",
-    "algebra/boolean.md": "56d2e0be2853d49b5dface7fa2d785a9",
+    "algebra/set.md": "92de8c0197cef54b5c6c269aac6b93b2",
+    "algebra/boolean.md": "ee41e624f4d3d3aca00020d9a9ae42c8",
    "git/merge-conflicts.md": "af3603c7a8fde60a1982c0950d209b77",
    "_journal/2024-05-28.md": "0f6aeb5ec126560acdc2d8c5c6570337",
    "_journal/2024-05/2024-05-27.md": "e498d5154558ebcf7261302403ea8016",
@ -489,10 +490,17 @@
    "_journal/2024-05/2024-05-31.md": "4421312bffa6f3c7e501a28be1e3bd5b",
    "_journal/2024-06-02.md": "0059e0332794e0007545b92bc4c1ff9f",
    "_journal/2024-06/2024-06-01.md": "6d39286db9d89975e441c57f6a92cfaf",
-    "_journal/2024-06-03.md": "560019592d88237bdb4b0ea609465ef6",
+    "_journal/2024-06-03.md": "3dbe96317f721515f3e6a20b82e3d537",
    "_journal/2024-06/2024-06-02.md": "0059e0332794e0007545b92bc4c1ff9f",
    "programming/lean.md": "2815eac192c7e231937e2369817a6dc1",
-    "logic/equality.md": "1867b676e0e7c9ec8f1addc40fefb966"
+    "logic/equality.md": "1867b676e0e7c9ec8f1addc40fefb966",
+    "_journal/2024-06-04.md": "a7ca58741dbfdd7e351d835db3884c68",
+    "_journal/2024-06/2024-06-03.md": "3dbe96317f721515f3e6a20b82e3d537",
+    "_journal/2024-06-05.md": "60f82c089f6911db8541a0bca7505ab4",
+    "_journal/2024-06/2024-06-04.md": "52b28035b9c91c9b14cef1154c1a0fa1",
+    "_journal/2024-06-06.md": "3f9109925dea304e7172df39922cc95a",
+    "_journal/2024-06/2024-06-05.md": "b06a0fa567bd81e3b593f7e1838f9de1",
+    "set/relations.md": "411e81682aa5a731cd501031dc3ab59d"
  },
  "fields_dict": {
    "Basic": [
--- a/notes/_journal/2024-06-06.md
+++ b/notes/_journal/2024-06-06.md
@ -0,0 +1,11 @@
+---
+title: "2024-06-06"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[relations#Overview|ordered pairs]].
--- a/notes/_journal/2024-06/2024-06-03.md
+++ b/notes/_journal/2024-06/2024-06-03.md
--- a/notes/_journal/2024-06/2024-06-04.md
+++ b/notes/_journal/2024-06/2024-06-04.md
@ -0,0 +1,12 @@
+---
+title: "2024-06-04"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[set#Symmetric Difference|symmetric differences]] of sets.
+* Finished Chapter 2 exercises in "Elements of Set Theory".
--- a/notes/_journal/2024-06/2024-06-05.md
+++ b/notes/_journal/2024-06/2024-06-05.md
@ -0,0 +1,11 @@
+---
+title: "2024-06-05"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Read chapter 18.10.2 and 15.4 in "Database System Concepts".
--- a/notes/algebra/boolean.md
+++ b/notes/algebra/boolean.md
@ -180,22 +180,55 @@ END%%

 %%ANKI
 Basic
-What C bit-level operator corresponds to $\cup$?
+What C bit-level operator corresponds to set notation $\cup$?
 Back: `|`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: binary c17 set
 <!--ID: 1707774068186-->
 END%%

+%%ANKI
+Cloze
+{$\cup$} is to the algebra of sets whereas {$\lor$} is to boolean algebra.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+Tags: set
+<!--ID: 1717554445676-->
+END%%
+
 %%ANKI
 Basic
-What C bit-level operator corresponds to $\cap$?
+What C bit-level operator corresponds to set notation $\cap$?
 Back: `&`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: binary c17 set
 <!--ID: 1707774068192-->
 END%%

+%%ANKI
+Cloze
+{$\cap$} is to the algebra of sets whereas {$\land$} is to boolean algebra.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+Tags: set
+<!--ID: 1717554445682-->
+END%%
+
+%%ANKI
+Basic
+What C bit-level operator corresponds to set notation $\triangle$?
+Back: `^`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: binary c17 set
+<!--ID: 1717554445689-->
+END%%
+
+%%ANKI
+Cloze
+{$\triangle$} is to the algebra of sets whereas {$\oplus$} is to boolean algebra.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+Tags: set
+<!--ID: 1717554445695-->
+END%%
+
 %%ANKI
 Basic
 What is a bit mask?
--- a/notes/algebra/set.md
+++ b/notes/algebra/set.md
@ -520,6 +520,34 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1717073537007-->
 END%%

+## Symmetric Difference
+
+Define the **symmetric difference** of sets $A$ and $B$ as $$A \mathop{\triangle} B = (A - B)  \cup (B - A)$$
+
+%%ANKI
+Basic
+What two operators are used in the definition of the symmetric difference?
+Back: $\cup$ and $-$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717554445662-->
+END%%
+
+%%ANKI
+Basic
+How is the symmetric difference of sets $A$ and $B$ denoted?
+Back: $A \mathop{\triangle} B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717554445665-->
+END%%
+
+%%ANKI
+Basic
+How is $A \mathop{\triangle} B$ defined?
+Back: As $(A - B)  \cup (B - A)$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717554445670-->
+END%%
+
 ## Bibliography

 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
--- a/notes/programming/lambda-calculus.md
+++ b/notes/programming/lambda-calculus.md
@ -16,6 +16,14 @@ Assume that there is given an infinite sequence of expressions called **variable

 If the sequence of atomic constants is empty, the system is called **pure**. Otherwise it is called **applied**.

+%%ANKI
+Basic
+Who is usually attributed the creation of $\lambda$-calculus?
+Back: Alonzo Church.
+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: 1717450542692-->
+END%%
+
 %%ANKI
 Basic
 What does a "higher-order function" refer to?
@ -769,6 +777,84 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
 <!--ID: 1717036717102-->
 END%%

+For all $\lambda$-terms $M$, $N$, and variables $x$:
+
+* $[x/x]M \equiv M$
+* $x \not\in FV(M) \Rightarrow [N/x]M \equiv M$
+* $x \in FV(M) \Rightarrow FV([N/x]M) = FV(N) \cup (FV(M) - \{x\})$
+
+%%ANKI
+Basic
+What is the result of $[x/x]M$?
+Back: $M$.
+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: 1717439837468-->
+END%%
+
+%%ANKI
+Basic
+If $x \not\in FV(M)$, what is the result of $[N/x]M$?
+Back: $M$
+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: 1717439837499-->
+END%%
+
+%%ANKI
+Basic
+Suppose $x \in FV(M)$. How is $FV([N/x]M)$ equivalently written without substitution?
+Back: $FV(N) \cup (FV(M) - \{x\})$
+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: 1717449967215-->
+END%%
+
+%%ANKI
+Basic
+Suppose $x \in FV(M)$. How is $FV(N) \cup (FV(M) - \{x\})$ more simply written using substitution?
+Back: $FV([N/x]M)$
+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: 1717449967220-->
+END%%
+
+%%ANKI
+Basic
+What is the result of $lgh([y/x]M)$?
+Back: $lgh(M)$
+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: 1717439837513-->
+END%%
+
+%%ANKI
+Basic
+$[N/x]M$ corresponds to which equivalence-transformation inference rule?
+Back: Substitution.
+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: 1717449830572-->
+END%%
+
+%%ANKI
+Basic
+$[P/v][v/x]M \equiv [P/x]M$ corresponds to which equivalence-transformation inference rule?
+Back: Transitivity.
+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: 1717449830601-->
+END%%
+
+%%ANKI
+Basic
+Rewrite $(E_u^x)_v^x$ using $\lambda$-calculus syntax.
+Back: $[v/x][u/x]E$
+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: 1717449830608-->
+END%%
+
+%%ANKI
+Basic
+Rewrite $[x/v][v/x]M$ using equivalence-transformation syntax.
+Back: $(M^x_v)^v_x$
+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: 1717449830614-->
+END%%
+
 ## Bibliography

 * 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).
--- a/notes/set/index.md
+++ b/notes/set/index.md
@ -468,7 +468,7 @@ END%%

 %%ANKI
 Basic
-What advantage does the general form of the union axiom have over its prelimiary form?
+What advantage does the general form of the union axiom have over its preliminary form?
 Back: The general form can handle infinite sets.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1716309007851-->
@ -746,7 +746,7 @@ END%%

 %%ANKI
 Basic
-What is the result of $\bigcap \{\{2, 4, 6\}, \{6, 16, 26\}, \{0\}\}$?
+What is the result of $\bigcap \{\{2, 4, 6\}, \{6, 16, 26\}, \{6\}\}$?
 Back: $\{6\}$
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1716309007870-->
@ -809,6 +809,15 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1716395245875-->
 END%%

+%%ANKI
+Basic
+What set operation is shaded green in the following venn diagram?
+![[venn-diagram-symm-diff.png]]
+Back: $A \mathop{\triangle} B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717554445655-->
+END%%
+
 %%ANKI
 Basic
 The "subset axioms" are more accurately classified as what?
--- a/notes/set/relations.md
+++ b/notes/set/relations.md
@ -0,0 +1,149 @@
+---
+title: Relations
+TARGET DECK: Obsidian::STEM
+FILE TAGS: set::relation
+tags:
+  - relation
+  - set
+---
+
+## Overview
+
+An ordered pair of $x$ and $y$, denoted $\langle x, y \rangle$, is defined as: $\langle x, y \rangle = \{\{x\}, \{x, y\}\}$. We define the **first coordinate** of $\langle x, y \rangle$ to be $x$ and the **second coordinate** to be $y$.
+
+%%ANKI
+Basic
+How is an ordered pair of $x$ and $y$ denoted?
+Back: $\langle x, y \rangle$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717678753102-->
+END%%
+
+%%ANKI
+Basic
+What property must any satisfactory definition of $\langle x, y \rangle$ satisfy?
+Back: $x$ and $y$, along with their order, are uniquely determined.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679524930-->
+END%%
+
+%%ANKI
+Basic
+Which of ordered pairs or sets is more general?
+Back: Sets.
+<!--ID: 1717678753108-->
+END%%
+
+%%ANKI
+Basic
+What biconditional is used to prove the well-definedness of $\langle x, y \rangle$?
+Back: $(\langle x, y \rangle = \langle u, v \rangle) \Leftrightarrow (x = u \land y = v)$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717678753111-->
+END%%
+
+%%ANKI
+Cloze
+{$\{1, 2\}$} is a set whereas {$\langle 1, 2 \rangle$} is an ordered pair.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717678753116-->
+END%%
+
+%%ANKI
+Basic
+How is $\langle x, y \rangle$ usually defined?
+Back: As $\{\{x\}, \{x, y\}\}$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717678753120-->
+END%%
+
+%%ANKI
+Basic
+Who is usually attributed the most commonly used definition of an ordered pair?
+Back: Kazimierz Kuratowski.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717678753124-->
+END%%
+
+%%ANKI
+Basic
+How is $\{\{x\}, \{x, y\}\}$ alternatively denoted?
+Back: $\langle x, y \rangle$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717678753129-->
+END%%
+
+%%ANKI
+Cloze
+Well-definedness of ordered pairs: {$\langle u, v \rangle = \langle x, y \rangle$} if and only if {$u = x \land v = y$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717678753134-->
+END%%
+
+%%ANKI
+Basic
+What term is used to refer to $x$ in $\langle x, y \rangle$?
+Back: The first coordinate.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717678753139-->
+END%%
+
+%%ANKI
+Cloze
+$y$ is the {second} coordinate of $\langle x, y \rangle$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717678753145-->
+END%%
+
+Given two sets $A$ and $B$, the **Cartesian product** $A \times B$ is defined as: $$A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$$
+
+%%ANKI
+Basic
+How is the Cartesian product of $A$ and $B$ denoted?
+Back: $A \times B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397781-->
+END%%
+
+%%ANKI
+Basic
+Using ordered pairs, how is $A \times B$ defined?
+Back: $\{\langle x, y \rangle \mid x \in A \land y \in B\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397797-->
+END%%
+
+%%ANKI
+Basic
+Who is attributed the representation of points in a plane?
+Back: René Descartes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397825-->
+END%%
+
+%%ANKI
+Basic
+Why is the Cartesian product named the way it is?
+Back: It is named after René Descartes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397836-->
+END%%
+
+%%ANKI
+Basic
+Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
+Back: $\mathscr{P}\mathscr{P}A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397848-->
+END%%
+
+%%ANKI
+Cloze
+{$x \in A$} iff {$\{x\} \subseteq A$} iff {$\{x\} \in \mathscr{P}A$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397860-->
+END%%
+
+## Bibliography
+
+* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
--- a/notes/x86-64/instructions.md
+++ b/notes/x86-64/instructions.md
@ -82,14 +82,6 @@ There are three types of operands:
 | Memory    | $(r_b,r_i,s)$     | $M[R[r_b] + R[r_i] \cdot s]$       | Scaled indexed      |
 | Memory    | $Imm(r_b,r_i,s)$  | $M[Imm + R[r_b] + R[r_i] \cdot s]$ | Scaled indexed      |

-%%ANKI
-Basic
-What are the three types of operands instructions can act on?
-Back: Immediates, registers, and memory addresses.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1713212889877-->
-END%%
-
 %%ANKI
 Basic
 What are the types of source operands instructions can specify?