r
/
notebook
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Notes on equality and formal systems.

c-declarations
Joshua Potter 2024-06-03 07:55:29 -06:00
parent 216ca47217
commit 9c39cc2b49
19 changed files with 557 additions and 31 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

34
notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
View File

@ -164,7 +164,7 @@
"posix/regexp.md": "887711b8fc64ca258d9987133ecf056c",
"journal/2024-02-04.md": "e2b5678fc53d7284b71ed6820c02b954",
"gawk/regexp.md": "d9229f1eabe1b99e965eecaa03bee86c",
"_templates/daily.md": "bfcd221502a23a285af100317b6ef127",
"_templates/daily.md": "36d0b695e8a955ff1d8e980154469e03",
"_journal/2024-02-05.md": "f8505abd415c50fd97c81fd6153a6d4f",
"_journal/2024-02-06.md": "1ea415f3c3f5be17f796b9a0d4df565f",
"_journal/2024-02-04.md": "f77a3c5f3ce7969120f226738836dc92",
@ -177,7 +177,7 @@
"algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
"algorithms/loop-invariant.md": "3b390e720f3b2a98e611b49a0bb1f5a9",
"algorithms/running-time.md": "5efc0791097d2c996f931c9046c95f65",
"algorithms/order-growth.md": "12bf6c10653912283921dcc46c7fa0f8",
"algorithms/order-growth.md": "4b03586ce9d75579f0a85be85445c1c4",
"_journal/2024-02-08.md": "19092bdfe378f31e2774f20d6afbfbac",
"algorithms/sorting/selection-sort.md": "73415c44d6f4429f43c366078fd4bf98",
"algorithms/index 1.md": "6fada1f3d5d3af64687719eb465a5b97",
@ -245,7 +245,7 @@
"_journal/2024-02/2024-02-21.md": "f423137ae550eb958378750d1f5e98c7",
"_journal/2024-02-23.md": "219ce9ad15a8733edd476c97628b71fd",
"_journal/2024-02/2024-02-22.md": "312e55d57868026f6e80f7989a889c2b",
"c17/strings.md": "192fe7fe4e321262dcd9ad2a1d322d8e",
"c17/strings.md": "c4da68e9280cda7bb5200382f83f6f6f",
"c17/index.md": "78576ee41d0185df82c59999142f4edb",
"c17/escape-sequences.md": "a8b99070336878b4e8c11e9e4525a500",
"c17/declarations.md": "65d2faaf8cc50878ad80126109e98b1f",
@ -313,7 +313,7 @@
"_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f",
"_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b",
"set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53",
"set/index.md": "74b50d8ca249c4e99b22abd4adedb300",
"set/index.md": "55a80d53973af2c0343438346e2086b2",
"set/graphs.md": "4bbcea8f5711b1ae26ed0026a4a69800",
"_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb",
"_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391",
@ -436,7 +436,7 @@
"_journal/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b",
"_journal/2024-05/2024-05-12.md": "ca9f3996272152ef89924bb328efd365",
"git/remotes.md": "2208e34b3195b6f1ec041024a66fb38b",
"programming/pred-trans.md": "6925b0be2de73c214df916a53e23fd5a",
"programming/pred-trans.md": "293806a24c2b9ef3fdb95ad74f9239b4",
"set/axioms.md": "063955bf19c703e9ad23be2aee4f1ab7",
"_journal/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5",
"_journal/2024-05/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b",
@ -451,8 +451,8 @@
"_journal/2024-05/2024-05-16.md": "9fdfadc3f9ea6a4418fd0e7066d6b10c",
"_journal/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c",
"hashing/direct-addressing.md": "87d1052ac7eae3061d88d011432cb693",
"hashing/index.md": "c870cf66e0224db58315ac0ba43b9cb1",
"set/classes.md": "18a09731868070e0c24a42bf0f582619",
"hashing/index.md": "8912a564ac1066de14044421fdc35b75",
"set/classes.md": "6776b4dc415021e0ef60b323b5c2d436",
"_journal/2024-05-19.md": "fddd90fae08fab9bd83b0ef5d362c93a",
"_journal/2024-05/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c",
"_journal/2024-05/2024-05-17.md": "fb880d68077b655ede36d994554f3aba",
@ -466,7 +466,7 @@
"programming/λ-Calculus.md": "bf36bdaf85abffd171bb2087fb8228b2",
"_journal/2024-05-23.md": "9d9106a68197adcee42cd19c69d2f840",
"_journal/2024-05/2024-05-22.md": "3c29eec25f640183b0be365e7a023750",
"programming/lambda-calculus.md": "151797fa3c012f1971d6a33eeea19274",
"programming/lambda-calculus.md": "547bc98111e516fc70a415d91db7b3ff",
"_journal/2024-05-25.md": "04e8e1cf4bfdbfb286effed40b09c900",
"_journal/2024-05/2024-05-24.md": "86132f18c7a27ebc7a3e4a07f4867858",
"_journal/2024-05/2024-05-23.md": "d0c98b484b1def3a9fd7262dcf2050ad",
@ -474,13 +474,25 @@
"_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": "d7b4c7943f3674bb152389f4bef1a234",
"algebra/set.md": "049c5ff36ce8fe99b2c8a8f452d0514f",
"algebra/boolean.md": "56d2e0be2853d49b5dface7fa2d785a9",
"git/merge-conflicts.md": "cb7d4d373639f75f6647be60f3fe97f3",
"git/merge-conflicts.md": "af3603c7a8fde60a1982c0950d209b77",
"_journal/2024-05-28.md": "0f6aeb5ec126560acdc2d8c5c6570337",
"_journal/2024-05/2024-05-27.md": "e498d5154558ebcf7261302403ea8016",
"_journal/2024-05-29.md": "aee3f3766659789d7dfb63dd247844cc",
"_journal/2024-05/2024-05-28.md": "28297d2a418f591ebc15c74fa459ddd9"
"_journal/2024-05/2024-05-28.md": "28297d2a418f591ebc15c74fa459ddd9",
"_journal/2024-05-30.md": "0467b8ded9cf1d9e3b419fa76f18f3c5",
"_journal/2024-05/2024-05-29.md": "ac77a3dfa0d0a538e7febd81886aba80",
"_journal/2024-05-31.md": "93b6f1e2d4a03406d8e738091874fbe4",
"_journal/2024-05/2024-05-30.md": "0b93c5b5e07130fac0e200754d68a450",
"_journal/2024-06-01.md": "46c00cc59a2ae126ad54e9114ee97646",
"_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/2024-06/2024-06-02.md": "0059e0332794e0007545b92bc4c1ff9f",
"programming/lean.md": "2815eac192c7e231937e2369817a6dc1",
"logic/equality.md": "1867b676e0e7c9ec8f1addc40fefb966"
},
"fields_dict": {
"Basic": [

0
notes/_journal/2024-05-29.md → notes/_journal/2024-05/2024-05-29.md
View File

11
notes/_journal/2024-05/2024-05-30.md Normal file
View File

@ -0,0 +1,11 @@
---
title: "2024-05-30"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Notes on monotonicity and antimonotonicity properties of $\subseteq$.

9
notes/_journal/2024-05/2024-05-31.md Normal file
View File

@ -0,0 +1,9 @@
---
title: "2024-05-31"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)

11
notes/_journal/2024-06-03.md Normal file
View File

@ -0,0 +1,11 @@
---
title: "2024-06-03"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Add notes on [[lambda-calculus#Syntactic Identity|syntactic identity]]. Relate this concept to Lean's syntactic equality.

9
notes/_journal/2024-06/2024-06-01.md Normal file
View File

@ -0,0 +1,9 @@
---
title: "2024-06-01"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)

11
notes/_journal/2024-06/2024-06-02.md Normal file
View File

@ -0,0 +1,11 @@
---
title: "2024-06-02"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Generalization of distributive laws and De Morgan's laws.

2
notes/_templates/daily.md
View File

@ -4,6 +4,6 @@ title: "{{date:YYYY-MM-DD}}"
- [ ] Anki Flashcards
- [ ] KoL
- [ ] OGS
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)

332
notes/algebra/set.md
View File

@ -141,6 +141,118 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1716803270452-->
END%%
%%ANKI
Basic
What concept in set theory relates the algebra of sets to boolean algebra?
Back: Membership.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322271-->
END%%
%%ANKI
Basic
What two equalities relates $A \cup B$ with $a \lor b$?
Back: $a = (x \in A)$ and $b = (x \in B)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322264-->
END%%
%%ANKI
Basic
What two equalities relates $A \cap B$ with $a \land b$?
Back: $a = (x \in A)$ and $b = (x \in B)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322275-->
END%%
More generally, for any sets $A$ and $\mathscr{B}$, $$\begin{align*} A \cup \bigcap \mathscr{B} & = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}, \text{ for } \mathscr{B} \neq \varnothing \\ A \cap \bigcup \mathscr{B} & = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\} \end{align*}$$
%%ANKI
Basic
What is the generalization of identity $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$?
Back: $A \cap \bigcup \mathscr{B} = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717366846568-->
END%%
%%ANKI
Basic
What is the generalization of identity $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$?
Back: $A \cup \bigcap \mathscr{B} = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$ for $\mathscr{B} \neq \varnothing$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717366846580-->
END%%
%%ANKI
Cloze
Assuming $\mathscr{B} \neq \varnothing$, the distributive law states {$A \cup \bigcap \mathscr{B}$} $=$ {$\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717366846573-->
END%%
%%ANKI
Cloze
The distributive law states {$A \cap \bigcup \mathscr{B}$} $=$ {$\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717366846594-->
END%%
%%ANKI
Basic
How is set $\{A \cup X \mid X \in \mathscr{B}\}$ pronounced?
Back: The set of all $A \cup X$ such that $X \in \mathscr{B}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717367767303-->
END%%
%%ANKI
Basic
What is the specialization of identity $A \cap \bigcup \mathscr{B} = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$?
Back: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717367767308-->
END%%
%%ANKI
Basic
What is the specialization of identity $A \cup \bigcap \mathscr{B} = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$?
Back: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717367767311-->
END%%
%%ANKI
Basic
Does $\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B}$ gets larger?
Back: Larger.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322278-->
END%%
%%ANKI
Basic
Does $\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B}$ gets smaller?
Back: Smaller.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322281-->
END%%
%%ANKI
Basic
Does $\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B} \neq \varnothing$ gets larger?
Back: Smaller.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322284-->
END%%
%%ANKI
Basic
Does $\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B} \neq \varnothing$ gets smaller?
Back: Larger.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322287-->
END%%
### De Morgan's Laws
For any sets $A$, $B$, and $C$, $$\begin{align*} C - (A \cup B) & = (C - A) \cap (C - B) \\ C - (A \cap B) & = (C - A) \cup (C - B) \end{align*}$$
@ -188,6 +300,226 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1716803270485-->
END%%
More generally, for any sets $C$ and $\mathscr{A} \neq \varnothing$, $$\begin{align*} C - \bigcup \mathscr{A} & = \bigcap\, \{C - X \mid X \in \mathscr{A}\} \\ C - \bigcap \mathscr{A} & = \bigcup\, \{C - X \mid X \in \mathscr{A}\} \end{align*}$$
%%ANKI
Basic
What is the generalization of identity $C - (A \cup B) = (C - A) \cap (C - B)$?
Back: $C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}$ for $\mathscr{A} \neq \varnothing$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717367767316-->
END%%
%%ANKI
Basic
What is the generalization of identity $C - (A \cap B) = (C - A) \cup (C - B)$?
Back: $C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}$ for $\mathscr{A} \neq \varnothing$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717367767323-->
END%%
%%ANKI
Cloze
For $\mathscr{A} \neq \varnothing$, De Morgan's law states that {$C - \bigcap \mathscr{A}$} $=$ {$\bigcup\, \{C - X \mid X \in \mathscr{A}\}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717367767320-->
END%%
%%ANKI
Basic
What is the specialization of identity $C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}$?
Back: $C - (A \cup B) = (C - A) \cap (C - B)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717373048517-->
END%%
%%ANKI
Basic
What is the specialization of identity $C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}$?
Back: $C - (A \cap B) = (C - A) \cup (C - B)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717373048522-->
END%%
%%ANKI
Basic
Which law of the algebra of sets is represented by e.g. $C - (A \cup B) = (C - A) \cap (C - B)$?
Back: De Morgan's Law.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717373048525-->
END%%
%%ANKI
Cloze
For $\mathscr{A} \neq \varnothing$, De Morgan's law states that {$C - \bigcup \mathscr{A}$} $=$ {$\bigcap\, \{C - X \mid X \in \mathscr{A}\}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717367767328-->
END%%
%%ANKI
Basic
Why does identity $C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}$ fail when $\mathscr{A} = \varnothing$?
Back: The RHS evaluates to class $\bigcap \varnothing$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717368301050-->
END%%
%%ANKI
Basic
Why does identity $C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}$ fail when $\mathscr{A} = \varnothing$?
Back: $\bigcap \mathscr{A}$ is undefined.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717368301055-->
END%%
%%ANKI
Basic
Does $\bigcap\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets larger?
Back: Smaller.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322295-->
END%%
%%ANKI
Basic
Does $\bigcap\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets smaller?
Back: Larger.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322299-->
END%%
%%ANKI
Basic
Does $\bigcup\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets larger?
Back: Larger.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372322304-->
END%%
%%ANKI
Does $\bigcup\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets smaller?
Back: Smaller.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
END%%
### Monotonicity
Let $A$, $B$, and $C$ be arbitrary sets. Then
* $A \subseteq B \Rightarrow A \cup C \subseteq B \cup C$,
* $A \subseteq B \Rightarrow A \cap C \subseteq B \cap C$,
* $A \subseteq B \Rightarrow \bigcup A \subseteq \bigcup B$
%%ANKI
Basic
What kind of propositional logical statement are the monotonicity properties of $\subseteq$?
Back: An implication.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536967-->
END%%
%%ANKI
Basic
What is the shared antecedent of the monotonicity properties of $\subseteq$?
Back: $A \subseteq B$ for some sets $A$ and $B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536973-->
END%%
%%ANKI
Basic
Given sets $A$, $B$, and $C$, state the monotonicity property of $\subseteq$ related to the $\cup$ operator.
Back: $A \subseteq B \Rightarrow A \cup C \subseteq B \cup C$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536976-->
END%%
%%ANKI
Basic
Given sets $A$, $B$, and $C$, state the monotonicity property of $\subseteq$ related to the $\cap$ operator.
Back: $A \subseteq B \Rightarrow A \cap C \subseteq B \cap C$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536979-->
END%%
%%ANKI
Basic
Given sets $A$ and $B$, state the monotonicity property of $\subseteq$ related to the $\bigcup$ operator.
Back: $A \subseteq B \Rightarrow \bigcup A \subseteq \bigcup B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536982-->
END%%
%%ANKI
Basic
Why are the monotonicity properties of $\subseteq$ named the way they are?
Back: The ordering of operands in the antecedent are preserved in the consequent.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536985-->
END%%
### Antimonotonicity
Let $A$, $B$, and $C$ be arbitrary sets. Then
* $A \subseteq B \Rightarrow C - B \subseteq C - A$,
* $\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A$
%%ANKI
Basic
What kind of propositional logical statement are the antimonotonicity properties of $\subseteq$?
Back: An implication.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536988-->
END%%
%%ANKI
Basic
What is the shared antecedent of the antimonotonicity properties of $\subseteq$?
Back: N/A.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536991-->
END%%
%%ANKI
Cloze
{1:Monotonicity} of $\subseteq$ is to {2:$\bigcup$} whereas {2:antimonotonicity} of $\subseteq$ is to {1:$\bigcap$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536994-->
END%%
%%ANKI
Basic
Why are the antimonotonicity properties of $\subseteq$ named the way they are?
Back: The ordering of operands in the antecedent are reversed in the consequent.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536998-->
END%%
%%ANKI
Basic
Given sets $A$ and $B$, state the antimonotonicity property of $\subseteq$ related to the $\bigcap$ operator.
Back: $\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073537001-->
END%%
%%ANKI
Basic
Given sets $A$, $B$, and $C$, state the antimonotonicity property of $\subseteq$ related to the $-$ operator.
Back: $A \subseteq B \Rightarrow C - B \subseteq C - A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073537004-->
END%%
%%ANKI
Basic
Why do we need the empty set check in $\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A$?
Back: $\bigcap A$ is not a set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073537007-->
END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

4
notes/algorithms/order-growth.md
View File

@ -13,8 +13,8 @@ The **running time** of an algorithm is usually considered as a function of its
%%ANKI
Basic
How is the running time of a program measured as a function?
Back: As a function of its input size.
What is the input of a function used to measure a program's running time?
Back: The size of the program's input.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1707334419352-->
END%%

4
notes/c17/strings.md
View File

@ -74,7 +74,7 @@ END%%
%%ANKI
Cloze
The {`width` and `precision`} fields are output related whereas the {`length`} field is input related.
The {1:`width`} and {1:`precision`} fields are output related whereas the {2:`length`} field is input related.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf
<!--ID: 1708425941269-->
@ -90,7 +90,7 @@ END%%
%%ANKI
Cloze
The {`-`} flag {left-aligns the output}.
The {`-`} flag {left-aligns} the output.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf
<!--ID: 1707918756812-->

2
notes/git/merge-conflicts.md
View File

@ -69,7 +69,7 @@ END%%
%%ANKI
Basic
In a `git merge`, what changes are between `=======` and `<<<<<<<`?
Back: The changes present on the branch being merged into `HEAD`.
Back: N/A.
Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014).
<!--ID: 1716804846996-->
END%%

2
notes/hashing/index.md
View File

@ -13,7 +13,7 @@ A **hash table** `T[0:m-1]` uses a **hash function** to map a universe of keys i
%%ANKI
Basic
With respect to hashing, what does the "universe" of keys refer to?
Back: Every potential key that may be inserted into the underlying dictionary.
Back: Every potential key that may be provided to the hash function.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716046153757-->
END%%

15
notes/logic/equality.md Normal file
View File

@ -0,0 +1,15 @@
---
title: Equality
TARGET DECK: Obsidian::STEM
FILE TAGS: equality
tags:
- equality
---
## Overview
## 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).

63
notes/programming/lambda-calculus.md
View File

@ -243,6 +243,53 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
<!--ID: 1716743248098-->
END%%
### Syntactic Identity
**Syntactic identity** of terms is denoted by "$\equiv$".
%%ANKI
Basic
What does it mean for two terms to be syntactically identical?
Back: The terms are written out using the exact same sequence of characters.
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: 1717422855675-->
END%%
%%ANKI
Basic
What form of Lean equality corresponds to $\lambda$-calculus's $\equiv$ operator?
Back: Syntactic equality.
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).
Tags: lean
<!--ID: 1717422855706-->
END%%
%%ANKI
Basic
How does Hindley et al. denote syntactic identity of $\lambda$-terms $M$ and $N$?
Back: $M \equiv N$
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: 1717422855711-->
END%%
%%ANKI
Basic
What syntactic identities are assumed when $MN \equiv PQ$?
Back: $M \equiv P$ and $N \equiv Q$.
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: 1717422855716-->
END%%
%%ANKI
Basic
What syntactic identities are assumed when $\lambda x. M \equiv \lambda y. P$?
Back: $x \equiv y$ and $M \equiv P$.
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: 1717422855722-->
END%%
### Length
The length of a $\lambda$-term (denoted $lgh$) is equal to the number of atoms in the term:
* $lgh(a) = 1$ for all atoms $a$;
@ -320,6 +367,8 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
<!--ID: 1716743248113-->
END%%
### Occurrence
For $\lambda$-terms $P$ and $Q$, the relation **$P$ occurs in $Q$** is defined by induction on $Q$ as:
* $P$ occurs in $P$;
@ -437,6 +486,8 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
<!--ID: 1716745016007-->
END%%
### Free and Bound Variables
An occurrence of a variable $x$ in a term $P$ is called
* **bound** if it is in the scope of a $\lambda x$ in $P$;
@ -574,7 +625,7 @@ END%%
%%ANKI
Basic
Which specific occurrences are bpund and binding in $\lambda x. x(\lambda y. yz)$?
Which specific occurrences are bound and binding in $\lambda x. x(\lambda y. yz)$?
Back: The first $x$ and the first $y$.
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: 1716745016031-->
@ -596,7 +647,7 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
<!--ID: 1716745016034-->
END%%
## Substitution
### Substitution
For any $M$, $N$, and $x$, define $[N/x]M$ to be the result of substituting $N$ for every free occurrence of $x$ in $M$, and changing bound variables to avoid clashes.
@ -634,9 +685,9 @@ END%%
%%ANKI
Cloze
$[N/x]M$ is the result of substituting {$N$} for every free occurrence of {$x$} in {$M$}.
$[N/x]M$ is the result of substituting {1:$N$} for every free occurrence of {1:$x$} in {1:$M$}.
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: 1717036717054-->
<!--ID: 1717251249627-->
END%%
%%ANKI
@ -656,7 +707,7 @@ END%%
%%ANKI
Basic
What is the result of $[N/x]a$, for some atom $a \neq x$?
What is the result of $[N/x]a$, for some atom $a \not\equiv x$?
Back: $a$
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: 1717036717064-->
@ -664,7 +715,7 @@ END%%
%%ANKI
Basic
What is the result of $[N/x]a$, for some atom $a = x$?
What is the result of $[N/x]a$, for some atom $a \equiv x$?
Back: $N$
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: 1717036717069-->

9
notes/programming/lean.md Normal file
View File

@ -0,0 +1,9 @@
---
title: Lean
TARGET DECK: Obsidian::STEM
FILE TAGS: lean
tags:
- lean
---
## Overview

10
notes/programming/pred-trans.md
View File

@ -563,14 +563,14 @@ END%%
%%ANKI
Basic
*Why* can't the $abort$ command be executed?
Back: Because it terminates in state $F$ which is impossible.
Back: By definition it executes in state $F$ which is impossible.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300129-->
END%%
%%ANKI
Basic
Which command does Gries introduce as the only whose predicate transformer is "constant"?
Which command does Gries introduce as the only "constant" predicate transformer?
Back: $abort$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300133-->
@ -578,15 +578,15 @@ END%%
%%ANKI
Basic
*Why* is $abort$ considered the only "constant" predicate transformer?
Back: The Law of the Excluded Miracle ensures $wp(S, F) = F$ for any other commands $S$.
How do we prove that $abort$ is the only "constant" predicate transformer?
Back: For any command $S$, the Law of the Excluded Miracle proves $wp(S, F) = F$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300137-->
END%%
%%ANKI
Basic
Consider $makeTrue$ defined as $wp(makeTrue, R) = T$ for all predicates $R$. What's wrong?
Suppose $makeTrue$ is defined as $wp(makeTrue, R) = T$ for all predicates $R$. What's wrong?
Back: If $R = F$, $makeTrue$ violates the Law of the Excluded Miracle.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300145-->

2
notes/set/classes.md
View File

@ -305,7 +305,7 @@ END%%
%%ANKI
Basic
Let $A$ be a set. What does $\{x \in A \mid x \not\in x\}$ evaluate to?
Back: $A$.
Back: $A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736492-->
END%%

58
notes/set/index.md
View File

@ -8,6 +8,30 @@ tags:
## Overview
%%ANKI
Basic
What are the two primitive notions of set theory?
Back: Sets and membership.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717417781230-->
END%%
%%ANKI
Basic
How does Enderton describe a primitive notion?
Back: An undefined concept other concepts are defined with.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717417781236-->
END%%
%%ANKI
Basic
Axioms can be thought of as doing what to primitive notions?
Back: Divulging partial information about their meaning.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717417781239-->
END%%
%%ANKI
Basic
How does Knuth define a *dynamic* set?
@ -223,6 +247,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1715649069259-->
END%%
%%ANKI
Basic
What axiom is used to prove two sets are equal to one another?
Back: Extensionality.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717372494462-->
END%%
## Empty Set Axiom
There exists a set having no members: $$\exists B, \forall x, x \not\in B$$
@ -747,7 +779,7 @@ END%%
%%ANKI
Basic
*Why* does $\bigcap \varnothing$ present a problem?
Back: Every set $x$ is a member of every member of $\varnothing$ (vacuously).
Back: Every set is a member of every member of $\varnothing$ (vacuously).
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007878-->
END%%
@ -777,6 +809,30 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1716395245875-->
END%%
%%ANKI
Basic
The "subset axioms" are more accurately classified as what?
Back: An axiom schema.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717368558153-->
END%%
%%ANKI
Basic
What is an axiom schema?
Back: An infinite bundle of axioms.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717368558159-->
END%%
%%ANKI
Basic
Which of the set theory axioms are more accurately described as an axiom schema?
Back: The subset axioms.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717368558164-->
END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).