Inductive sets, `enum` types, and BFS.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -194,7 +194,17 @@
     "function-kernel.png",
     "peano-system-i.png",
     "peano-system-ii.png",
-    "relation-ordering-example.png"
+    "relation-ordering-example.png",
+    "archimedean-property.png",
+    "church-rosser.png",
+    "directed-graph-example.png",
+    "undirected-graph-example.png",
+    "cyclic-undirected-labelled.png",
+    "graph-isomorphic.png",
+    "graph-induced-subgraph.png",
+    "graph-subgraph.png",
+    "graph-non-subgraph.png",
+    "bfs.gif"
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   "File Hashes": {
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@@ -374,8 +384,8 @@
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     "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53",
-    "set/index.md": "7d09418b46856b721f14c5c1bc7320fa",
+    "set/index.md": "9ac07f7fc1fadf1b42278df90b464adb",
-    "set/graphs.md": "15aa43bf7f73347219f822e4b400e9bf",
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@@ -568,8 +578,8 @@
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-    "x86-64/instructions/logical.md": "818428b9ef84753920dc61e5c2de9199",
+    "x86-64/instructions/logical.md": "49d40018f1fcb4ed1595d9175bbaab57",
-    "x86-64/instructions/arithmetic.md": "271218d855e7291f119f96e91f582738",
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-    "set/functions.md": "59e449d6756b57c846cdf07b0a1b4330",
+    "set/functions.md": "b93f460500a6a7228607f842636ed3b3",
     "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4",
     "_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307",
-    "lambda-calculus/beta-reduction.md": "a8e2825c84e842ceef7aa638a493b91a",
+    "lambda-calculus/beta-reduction.md": "0935987f2bac0e6298735f2b26fd5885",
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@@ -675,7 +685,7 @@
     "logic/classical/index.md": "ee0a4b2bfcfa2cab0880db449cb62df1",
     "logic/classical/truth-tables.md": "b739e2824a4a5c26ac446e7c15ce02aa",
     "formal-system/proof-system/index.md": "800e93b72a9852ea4823ab0a40854bba",
-    "formal-system/proof-system/equiv-trans.md": "47afc3ffa0bb758d9629d9d4a401394e",
+    "formal-system/proof-system/equiv-trans.md": "e2eae52f49249b622b87c7fd06967666",
     "formal-system/logical-system/index.md": "708bb1547e7343c236068c18da3f5dc0",
     "formal-system/logical-system/pred-logic.md": "34e872f4f920bf4e8c2cd00ee95b310f",
     "formal-system/logical-system/prop-logic.md": "b61ce051795d5a951c763b928ec5cea8",
@@ -746,7 +756,7 @@
     "_journal/2024-08/2024-08-15.md": "7c3a96a25643b62b0064bf32cb17d92f",
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     "_journal/2024-08/2024-08-16.md": "da1127a1985074a3930b4c3512344025",
-    "set/order.md": "b69f922200514975b7a7028eef030b59",
+    "set/order.md": "3bf63dd9c8ce6d2b4c6905dab0bd4aad",
     "_journal/2024-08-18.md": "6f8aec69e00401b611db2a377a3aace5",
     "ontology/philosophy/properties.md": "41b32249d3e4c23d73ddb3a417d65a4c",
     "_journal/2024-08-19.md": "94836e52ec04a72d3e1dbf3854208f65",
@@ -756,14 +766,14 @@
     "calculus/index.md": "5ee4d950533ae330ca5ef9e113fe87f3",
     "x86-64/instructions/conditions.md": "c5571deac40ac2eeb8666f2d3b3c278e",
     "_journal/2024-08-20.md": "e8bec308d1b29e411c6799ace7ef6571",
-    "algebra/arch-prop.md": "eccdd685f12898ed8679b558d19dc20a",
+    "algebra/arch-prop.md": "bca3724ef5aae3f7f20907108087af47",
     "_journal/2024-08/2024-08-19.md": "94836e52ec04a72d3e1dbf3854208f65",
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     "_journal/2024-08-23.md": "3b2feab2cc927e267263cb1e9c173d50",
-    "set/natural-numbers.md": "353d1eef7b50fa8f635adc928df734fa",
+    "set/natural-numbers.md": "a14465985e87b81ccbe3a5d27159654a",
     "_journal/2024-08-24.md": "563fad24740e44734a87d7c3ec46cec4",
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     "_journal/2024-08/2024-08-22.md": "050235d5dc772b542773743b57ce3afe",
@@ -771,7 +781,7 @@
     "c17/enums.md": "9414fb67aa256a0a11b7240534c67bf6",
     "c17/derived-types.md": "6fb8f23a2423f05d5bdccb6672a32e38",
     "c17/basic-types.md": "7c6653bf6dc24c2f2aa72fc95c4f7875",
-    "c17/types/simple.md": "36445dec496b5f7a066bdb7738b2f17e",
+    "c17/types/simple.md": "0fab7afc623e441cd17ae23497f6babe",
     "c17/types/enumerated.md": "e1f70a30677c776b7b44ac3e0ff4e76d",
     "c17/types/derived.md": "aff0d2b6d218fb67af3cc92ead924de3",
     "c17/types/basic.md": "5064e21e683c0218890058882e06b6f3",
@@ -784,7 +794,7 @@
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     "_journal/2024-08/2024-08-26.md": "6f40716e2f01cd097d4881259babf1ba",
-    "c17/types/conversions.md": "477528bf1a297a8fc4eed0ecb4206158",
+    "c17/types/conversions.md": "843ece1b41a227618c797b25cbcdd98b",
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@@ -816,7 +826,14 @@
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-    "_journal/2024-09-19.md": "612f08e8f9ce35f71378e2c4a636c862"
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+    "_journal/2024-09-22.md": "2d00b00b4eb9964465f30210187603cf",
+    "_journal/2024-09/2024-09-21.md": "2e6425f4db0187082947c3e0cb24f754",
+    "algorithms/bfs.md": "2734400c0a5843c92adbc2a0f06d4d56"
   },
   "fields_dict": {
     "Basic": [

notes/_journal/2024-09-22.md

@@ -0,0 +1,13 @@
+---
+title: "2024-09-22"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Finished "Inductive Sets" in chapter 4 of "Elements of Set Theory".
+* Finished notes on C's enumerated types.
+* Notes on [[bfs|BFS]].

notes/_journal/2024-09/2024-09-20.md

@@ -0,0 +1,9 @@
+---
+title: "2024-09-20"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/_journal/2024-09/2024-09-21.md

@@ -0,0 +1,11 @@
+---
+title: "2024-09-21"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Additional notes on transitive sets.

notes/algebra/arch-prop.md

@@ -78,7 +78,7 @@ END%%
 
 %%ANKI
 Basic
-What mathematical entities are assumed to exist in the formulate of the Archimedean property of the reals?
+What mathematical entities are presumed in 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-->

notes/algorithms/bfs.md

@@ -0,0 +1,177 @@
+---
+title: Breadth-First Search
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algorithm data_structure::graph
+tags:
+  - bfs
+  - graph
+---
+
+## Overview
+
+Bread-first search operates on a graph $G = \langle V, E \rangle$ and a **source** vertex $s$. It works by distinguishing between discovered and undiscovered nodes, incrementally marking nodes adjacent to discovered nodes from undiscovered to discovered.
+
+![[bfs.gif]]
+
+%%ANKI
+Basic
+What is BFS an acronym for?
+Back: **B**readth-**f**irst **s**earch.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727042295709-->
+END%%
+
+%%ANKI
+Cloze
+Breadth-first search is characterized by a graph and a {source vertex}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727042295717-->
+END%%
+
+%%ANKI
+Cloze
+The {1:source} of breadth-first {2:search} is the {2:root} of the breadth-first {1:tree}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727042295723-->
+END%%
+
+%%ANKI
+Basic
+Which of undirected and directed graphs is BFS applicable to?
+Back: Both.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727042295728-->
+END%%
+
+%%ANKI
+Basic
+With respect to breadth-first trees, what does the predecessor of a node $N$ refer to?
+Back: The node from which $N$ was discovered.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727042295733-->
+END%%
+
+%%ANKI
+Basic
+With respect to breadth-first trees, what does the parent of a node $N$ refer to?
+Back: The node from which $N$ was discovered.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727042295739-->
+END%%
+
+%%ANKI
+Basic
+With respect to breadth-first trees, the predecessor of a node is also known as what?
+Back: The parent of the node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035958-->
+END%%
+
+%%ANKI
+Basic
+With respect to breadth-first trees, the parent of a node is also known as what?
+Back: The predecessor of the node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035963-->
+END%%
+
+%%ANKI
+Basic
+What ADT is typically used to manage the set of most recently discovered BFS vertices?
+Back: A queue.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727042295745-->
+END%%
+
+%%ANKI
+Basic
+Which vertices are not discovered during a graph BFS?
+Back: Those not reachable from the source vertex.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035966-->
+END%%
+
+%%ANKI
+Basic
+What basic graph algorithm is the following a demonstration of?
+![[bfs.gif]]
+Back: Breadth-first search.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035969-->
+END%%
+
+%%ANKI
+Basic
+In BFS, what happens to the nodes found within the internal queue?
+Back: Undiscovered nodes adjacent to those in the queue are enqueued.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035972-->
+END%%
+
+%%ANKI
+Basic
+Which of BFS or DFS is used to find shortest paths?
+Back: BFS.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035975-->
+END%%
+
+%%ANKI
+Basic
+When can BFS *not* be used to find shortest paths?
+Back: When the graph in question has differently weighted edges.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035977-->
+END%%
+
+%%ANKI
+Basic
+Which weighted graphs can BFS be used on to find shortest paths?
+Back: Graphs with equally weighted edges.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035980-->
+END%%
+
+%%ANKI
+Basic
+Which standard graph representation has worst-case BFS running time of $O(\lvert V \rvert + \lvert E \rvert)$?
+Back: The adjacency-matrix representation.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035989-->
+END%%
+
+%%ANKI
+Basic
+Given graph $\langle V, E \rangle$ with adjacency-list representation, what is the worst-case run time of BFS?
+Back: $O(\lvert V \rvert + \lvert E \rvert)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035983-->
+END%%
+
+%%ANKI
+Basic
+Given graph $\langle V, E \rangle$ with adjacency-matrix representation, what is the worst-case run time of BFS?
+Back: $O(\lvert V \rvert^2)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044035986-->
+END%%
+
+%%ANKI
+Basic
+*Why* is BFS of an adjacency-list representation $O(\lvert V \rvert + \lvert E \rvert)$?
+Back: For each vertex being analyzed, we only examine its immediately adjacent vertices.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044184060-->
+END%%
+
+%%ANKI
+Basic
+*Why* is BFS of an adjacency-matrix representation $O(\lvert V \rvert^2)$?
+Back: For each vertex being analyzed, we must examine $\lvert V \rvert$ entries for adjacent vertices.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1727044184066-->
+END%%
+
+## Bibliography
+
+* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/c17/types/conversions.md

@@ -216,14 +216,26 @@ What is the type domain of `a + b`?
 unsigned short a;
 signed int b;
 ```
-Back: Indeterminate.
+Back: Real.
 Reference: “ISO: Programming Languages - C,” April 12, 2011, [https://port70.net/~nsz/c/c11/n1570.pdf](https://port70.net/~nsz/c/c11/n1570.pdf).
 <!--ID: 1724762203469-->
 END%%
 
 %%ANKI
 Basic
-*Why* is the type domain of `a + b` indeterminate?
+What is the common real type of `a + b`?
+```c
+unsigned short a;
+signed int b;
+```
+Back: Indeterminate.
+Reference: “ISO: Programming Languages - C,” April 12, 2011, [https://port70.net/~nsz/c/c11/n1570.pdf](https://port70.net/~nsz/c/c11/n1570.pdf).
+<!--ID: 1726840632804-->
+END%%
+
+%%ANKI
+Basic
+*Why* is the common real type of `a + b` indeterminate?
 ```c
 unsigned short a;
 int b;
@@ -235,7 +247,7 @@ END%%
 
 %%ANKI
 Basic
-What *might* the type domain of `a + b` be?
+What *might* the common real type of `a + b` be?
 ```c
 unsigned short a;
 signed int b;
@@ -259,7 +271,7 @@ END%%
 
 %%ANKI
 Basic
-When is the type domain of `a + b` equal to `unsigned int`?
+When is the common real type of `a + b` equal to `unsigned int`?
 ```c
 unsigned short a;
 signed int b;

notes/c17/types/simple.md

@@ -764,6 +764,64 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1723938382398-->
 END%%
 
+### Integer Constant Expressions
+
+An integer constant expression (ICE) is a compile-time integer value. Its value must be determinable at compile time (e.g. no function calls are permitted), and also no evaluation of an object must participate as an operand.
+
+%%ANKI
+Basic
+What is ICE an acronym for?
+Back: **I**nteger **c**onstant **e**xpression.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810784-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't `b42` in the following considered an ICE?
+```c
+enum { b42 = 42 }
+```
+Back: N/A. It is.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810792-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't `b42` in the following considered an ICE?
+```c
+signed const a42 = 42;
+enum { b42 = a42 }
+```
+Back: Because it depends on the evaluation of object `a42`.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810798-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't `c52` in the following considered an ICE?
+```c
+enum { b42 = 42, c52 = b42 + 10 }
+```
+Back: N/A. It is.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810804-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't `b42` in the following considered an ICE?
+```c
+signed const a42() { return 42; }
+enum { b42 = a42() }
+```
+Back: Because it depends on a function call.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810810-->
+END%%
+
 ## Floating Point
 
 ### Literals
@@ -933,7 +991,56 @@ Reference: Van der Linden, Peter. _Expert C Programming: Deep C Secrets_. Progra
 <!--ID: 1722786892138-->
 END%%
 
+%%ANKI
+Basic
+Positional values of `enum`s start at what value?
+Back: `0`
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810814-->
+END%%
+
+%%ANKI
+Basic
+What type is given to enumeration constants?
+Back: `signed int`
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810819-->
+END%%
+
+%%ANKI
+Basic
+What kind of integer expressions are `enum` constants limited to?
+Back: Integer constant expressions.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810827-->
+END%%
+
+%%ANKI
+Basic
+Besides being an ICE, what other condition is expected on `enum` values?
+Back: The ICE evaluates to an integer that fits within a `signed int`.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810833-->
+END%%
+
+%%ANKI
+Basic
+What mechanism(s) are available for defining constants of type `signed int`?
+Back: `enum`s and macros.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810839-->
+END%%
+
+%%ANKI
+Basic
+What mechanism(s) are available for defining constants of type `unsigned int`?
+Back: Macros.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727022810842-->
+END%%
+
 ## Bibliography
 
 * “ISO: Programming Languages - C,” April 12, 2011, [https://port70.net/~nsz/c/c11/n1570.pdf](https://port70.net/~nsz/c/c11/n1570.pdf).
+* Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
 * Van der Linden, Peter. _Expert C Programming: Deep C Secrets_. Programming Languages / C. Mountain View, Cal.: SunSoft Pr, 1994.

notes/formal-system/proof-system/equiv-trans.md

@@ -468,13 +468,6 @@ $$
 <!--ID: 1707253246457-->
 END%%
 
-%%ANKI
-Cloze
-The system of evaluation has {equivalences} whereas the formal system has {theorems}.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1707253246458-->
-END%%
-
 %%ANKI
 Basic
 What is a "theorem" in the equivalence-transformation formal system?

notes/lambda-calculus/beta-reduction.md

@@ -395,7 +395,7 @@ END%%
 
 %%ANKI
 Basic
-What proposition explains how applications relate to the definition of $\beta\text{-nf}$?
+What proposition explains how atoms and applications relate to the definition of $\beta\text{-nf}$?
 Back: For all atoms $a$, if $M, N \in \beta\text{-nf}$, then $aMN \in \beta\text{-nf}$.
 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: 1720645978924-->

notes/set/functions.md

@@ -1655,7 +1655,7 @@ END%%
 
 %%ANKI
 Basic
-Let $A$ be closed under $S$. With maximum specificity, what kind of mathematical object is $A$?
+Let $A$ be closed under $S$. What kind of mathematical object is $A$?
 Back: A set.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1726363069988-->
@@ -1663,7 +1663,7 @@ END%%
 
 %%ANKI
 Basic
-Let $A$ be closed under $S$. With maximum specificity, what kind of mathematical object is $S$?
+Let $A$ be closed under $S$. What kind of mathematical object is $S$?
 Back: A function.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1726363069991-->

notes/set/graphs.md

@@ -1203,7 +1203,7 @@ END%%
 
 %%ANKI
 Basic
-With maximum specificity, is $\langle B, D, E, J, K, B, A \rangle$ a path, trail, or walk?
+Is $\langle B, D, E, J, K, B, A \rangle$ most precisely a path, trail, or walk?
 ![[cyclic-undirected-labelled.png]]
 Back: A trail.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
@@ -1212,7 +1212,7 @@ END%%
 
 %%ANKI
 Basic
-With maximum specificity, is $\langle B, D, E, J, K, B \rangle$ a path, trail, or walk?
+Is $\langle B, D, E, J, K, B \rangle$ most precisely a path, trail, or walk?
 ![[cyclic-undirected-labelled.png]]
 Back: A path.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
@@ -1221,7 +1221,7 @@ END%%
 
 %%ANKI
 Basic
-With maximum specificity, is $\langle B, D, B, K, L \rangle$ a path, trail, or walk?
+Is $\langle B, D, B, K, L \rangle$ most precisely a path, trail, or walk?
 ![[cyclic-undirected-labelled.png]]
 Back: A walk.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
@@ -1230,7 +1230,7 @@ END%%
 
 %%ANKI
 Basic
-With maximum specificity, is $\langle A, B, D \rangle$ a path, trail, or walk?
+Is $\langle A, B, D \rangle$ most precisely a path, trail, or walk?
 ![[cyclic-undirected-labelled.png]]
 Back: A path.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).

notes/set/index.md

@@ -787,6 +787,38 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1715900348160-->
 END%%
 
+%%ANKI
+Basic
+Let $A$ be a set. What does $\bigcup \mathscr{P} A$ evaluate to?
+Back: $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726976526809-->
+END%%
+
+%%ANKI
+Basic
+Let $A$ be a set. *Why* does $\bigcup \mathscr{P} A = A$?
+Back: Because $\mathscr{P} A$ evaluates to the subsets of $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726976526815-->
+END%%
+
+%%ANKI
+Basic
+Let $A$ be a set. What does $\bigcap \mathscr{P} A$ evaluate to?
+Back: $\varnothing$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726976526819-->
+END%%
+
+%%ANKI
+Basic
+Let $A$ be a set. *Why* does $\bigcap \mathscr{P} A = \varnothing$?
+Back: Because $\varnothing \in \mathscr{P} A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726976526824-->
+END%%
+
 ## Subset Axioms
 
 For each formula $\_\_\_$ not containing $B$, the following is an axiom: $$\forall t_1, \cdots, \forall t_k, \forall c, \exists B, \forall x, (x \in B \Leftrightarrow x \in c \land \_\_\_)$$

notes/set/natural-numbers.md

@@ -283,7 +283,7 @@ END%%
 
 %%ANKI
 Basic
-Suppose $n \in \omega$. What other sets *must* $n$ be a member of?
+Suppose $n \in \omega$. By definition of natural numbers, 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-->
@@ -377,6 +377,8 @@ A **Peano system** is a triple $\langle N, S, e \rangle$ consisting of a set $N$
 * $S$ is one-to-one;
 * Any subset $A$ of $N$ that contains $e$ and is closed under $S$ equals $N$ itself.
 
+Given $\sigma = \{\langle n, n^+ \rangle \mid n \in \omega\}$, $\langle \omega, \sigma, 0 \rangle$ is a Peano system.
+
 %%ANKI
 Basic
 A Peano system is a tuple consisting of how many members?
@@ -387,7 +389,7 @@ END%%
 
 %%ANKI
 Basic
-Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $N$?
+Consider Peano system $\langle N, S, e \rangle$. What kind of mathematical object is $N$?
 Back: A set.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1726364667620-->
@@ -395,7 +397,7 @@ END%%
 
 %%ANKI
 Basic
-Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $S$?
+Consider Peano system $\langle N, S, e \rangle$. What kind of mathematical object is $S$?
 Back: A function.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1726364667623-->
@@ -419,7 +421,7 @@ END%%
 
 %%ANKI
 Basic
-Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $e$?
+Consider Peano system $\langle N, S, e \rangle$. What kind of mathematical object is $e$?
 Back: A set or urelement.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1726364667632-->
@@ -538,6 +540,70 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1726364667688-->
 END%%
 
+%%ANKI
+Basic
+Which Peano system serves as the prototypical example?
+Back: $\langle \omega, \sigma, 0 \rangle$ where $\sigma$ denotes the successor restricted to the natural numbers.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726928580006-->
+END%%
+
+%%ANKI
+Basic
+Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. How is $\omega$ defined?
+Back: As the set of natural numbers.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726928580037-->
+END%%
+
+%%ANKI
+Basic
+Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. How is $\sigma$ defined?
+Back: $\{\langle n, n^+ \rangle \mid n \in \omega\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726928580064-->
+END%%
+
+%%ANKI
+Basic
+Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. What kind of mathematical object is $\sigma$?
+Back: A function.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726928580069-->
+END%%
+
+%%ANKI
+Basic
+Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. What is the domain of $\sigma$?
+Back: $\omega$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726928580075-->
+END%%
+
+%%ANKI
+Basic
+Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. What is the codomain of $\sigma$?
+Back: $\omega$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726928580081-->
+END%%
+
+%%ANKI
+Basic
+Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. Its Peano induction postulate goes by what other name?
+Back: The induction principle for $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726928580087-->
+END%%
+
+%%ANKI
+Basic
+Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. The induction principle for $\omega$ satisfies what postulate of the system?
+Back: The Peano induction postulate.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726928580092-->
+END%%
+
 ## Transitivity
 
 A set $A$ is said to be **transitive** iff every member of a member of $A$ is itself a member of $A$. We can equivalently express this using any of the following formulations:
@@ -668,6 +734,110 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1726797209168-->
 END%%
 
+%%ANKI
+Basic
+Which sets serve as the prototypical example of transitive sets?
+Back: The natural numbers.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726857149204-->
+END%%
+
+%%ANKI
+Basic
+Is $n \in \omega$ a transitive set?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726857149214-->
+END%%
+
+%%ANKI
+Basic
+Is $\omega$ a transitive set?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726857149225-->
+END%%
+
+%%ANKI
+Basic
+How can we alternatively state "$\omega$ is a transitive set"?
+Back: Every natural number is a set of natural numbers.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726976055230-->
+END%%
+
+%%ANKI
+Basic
+How can we more concisely state "every natural number is a set of natural numbers"?
+Back: $\omega$ is a transitive set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726976055239-->
+END%%
+
+%%ANKI
+Basic
+What does $\mathscr{P}\,0$ evaluate to?
+Back: $1$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727019806525-->
+END%%
+
+%%ANKI
+Basic
+What does $\mathscr{P}\,1$ evaluate to?
+Back: $2$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727019806532-->
+END%%
+
+%%ANKI
+Basic
+What does $\mathscr{P}\,2$ evaluate to?
+Back: $\{0, 1, 2, \{1\}\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727019806534-->
+END%%
+
+%%ANKI
+Basic
+Suppose $X$ is transitive. Is $\bigcup X$ transitive?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727019806538-->
+END%%
+
+%%ANKI
+Basic
+Suppose for all $x \in X$, $x$ is transitive. Is $X$ transitive?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727019806541-->
+END%%
+
+%%ANKI
+Basic
+Suppose for all $x \in X$, $x$ is transitive. Is $\bigcup X$ transitive?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727019806545-->
+END%%
+
+%%ANKI
+Basic
+Suppose for all $x \in X$, $x$ is transitive. Is $\bigcap X$ transitive?
+Back: N/A. If $X = \varnothing$, $\bigcap X$ is undefined.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727019806550-->
+END%%
+
+%%ANKI
+Basic
+Suppose $X \neq \varnothing$ and for all $x \in X$, $x$ is transitive. Is $\bigcap X$ transitive?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727019806554-->
+END%%
+
 ## Bibliography
 
 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

notes/set/order.md

@@ -487,7 +487,7 @@ END%%
 
 %%ANKI
 Basic
-Consider an equivalence class of $x$ (modulo $R$). With maximum specificity, what kind of mathematical object is $R$?
+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-->

notes/x86-64/instructions/arithmetic.md

@@ -32,7 +32,7 @@ END%%
 
 %%ANKI
 Cloze
-The {`leaq`} instruction is to x86-64 as the {`&`} operator is to C.
+The {`leaq`} instruction is to x86-64 as the {`&`} unary operator is to C.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: c17
 <!--ID: 1715780601458-->

notes/x86-64/instructions/logical.md

@@ -31,7 +31,7 @@ END%%
 
 %%ANKI
 Cloze
-The {`NEG`} instruction class is to x86-64 whereas the {`-`} *unary* operator is to C.
+The {`NEG`} instruction class is to x86-64 whereas the {`-`} unary operator is to C.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: c17
 <!--ID: 1716126147801-->