From f08d90bed4a995e2f94d6b2a6aed4654222608d5 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Sat, 24 Aug 2024 02:07:51 -0600 Subject: [PATCH] Archimedean property and natural numbers. --- .../plugins/obsidian-to-anki-plugin/data.json | 34 ++- notes/_journal/2024-08-24.md | 11 + notes/_journal/{ => 2024-08}/2024-08-19.md | 0 notes/_journal/2024-08/2024-08-20.md | 11 + notes/_journal/2024-08/2024-08-21.md | 9 + notes/_journal/2024-08/2024-08-22.md | 11 + notes/_journal/2024-08/2024-08-23.md | 9 + notes/algebra/arch-prop.md | 98 ++++++ notes/algebra/set.md | 8 - notes/c17/index.md | 2 +- notes/c17/types.md | 2 +- .../calculus/images/archimedean-property.png | Bin 0 -> 3351 bytes notes/programming/pred-trans.md | 2 +- notes/set/index.md | 27 ++ notes/set/natural-numbers.md | 286 ++++++++++++++++++ notes/set/order.md | 20 +- notes/x86-64/registers.md | 16 +- 17 files changed, 507 insertions(+), 39 deletions(-) create mode 100644 notes/_journal/2024-08-24.md rename notes/_journal/{ => 2024-08}/2024-08-19.md (100%) create mode 100644 notes/_journal/2024-08/2024-08-20.md create mode 100644 notes/_journal/2024-08/2024-08-21.md create mode 100644 notes/_journal/2024-08/2024-08-22.md create mode 100644 notes/_journal/2024-08/2024-08-23.md create mode 100644 notes/algebra/arch-prop.md create mode 100644 notes/calculus/images/archimedean-property.png create mode 100644 notes/set/natural-numbers.md diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 40bc570..f38b3c7 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -195,7 +195,9 @@ "ordered-rooted-tree-cmp.png", "ordered-binary-tree-cmp.png", "lcrs-nodes.png", - "binary-tree-nodes.png" + "binary-tree-nodes.png", + "archimedean-property.png", + "infinite-cartesian-product.png" ], "File Hashes": { "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8", @@ -308,7 +310,7 @@ "_journal/2024-02-23.md": "219ce9ad15a8733edd476c97628b71fd", "_journal/2024-02/2024-02-22.md": "312e55d57868026f6e80f7989a889c2b", "c17/strings.md": "2da50edd26eae35c81f70e65bbd12d49", - "c17/index.md": "c96078cda31616017b0a6036ac87e60b", + "c17/index.md": "fd48bc8d8b9b28702e8fdf0f4cf977d5", "c17/escape-sequences.md": "a8b99070336878b4e8c11e9e4525a500", "c17/declarations.md": "2b61706906d8ae935e0b56e962ad2fa8", "algorithms/sorting/merge-sort.md": "6506483f7df6507cee0407bd205dbedd", @@ -375,7 +377,7 @@ "_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f", "_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b", "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53", - "set/index.md": "c103501e345a1b8201a26f2e83ed8379", + "set/index.md": "91060cf5e604f7683a34710dda2ea10b", "set/graphs.md": "6f08a3e8e4896b0325aef6c452bfbb56", "_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb", "_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391", @@ -498,11 +500,11 @@ "_journal/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b", "_journal/2024-05/2024-05-12.md": "ca9f3996272152ef89924bb328efd365", "git/remotes.md": "cbe2cd867f675f156e7fe71ec615890d", - "programming/pred-trans.md": "3c112418e7aa0970a9c38216a65b0932", + "programming/pred-trans.md": "c02471c6c9728dd19f8df7bc180ef8b1", "set/axioms.md": "063955bf19c703e9ad23be2aee4f1ab7", "_journal/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5", "_journal/2024-05/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b", - "x86-64/registers.md": "e55217fb711495490546975a7828e8f1", + "x86-64/registers.md": "5cb49ae47fb0f95df6e15991274f4ad3", "_journal/2024-05-15.md": "4e6a7e6df32e93f0d8a56bc76613d908", "_journal/2024-05/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5", "_journal/2024-05-16.md": "580c7ec61ec56be92fa8d6affcf0a5f6", @@ -536,7 +538,7 @@ "_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": "97ee8ac4f147ed64496b8a757265d1d9", + "algebra/set.md": "a89ada021de83240724adb70490e3472", "algebra/boolean.md": "fc47edb7d0080b73ce1ce0d3e0e16d7d", "git/merge-conflicts.md": "761ad6137ec51d3877f7d5b3615ca5cb", "_journal/2024-05-28.md": "0f6aeb5ec126560acdc2d8c5c6570337", @@ -737,7 +739,7 @@ "_journal/2024-08/2024-08-10.md": "08e7ea4a78c46645b93ec51e2372d04f", "_journal/2024-08-12.md": "8a37a2d1381f9d9e29d83031bad80dd0", "_journal/2024-08/2024-08-11.md": "acc91e07b43590e90846d2c936dcb3d5", - "c17/types.md": "5ff85d535ee99d3e7aa79da93eb8383c", + "c17/types.md": "069d0f7a38f5ae817945d5b6937dc1ec", "_journal/2024-08-14.md": "800650b9fa2f4445a174e0a547c2fa95", "_journal/2024-08/2024-08-13.md": "8b64225b06d1164a91176b123a3513a2", "_journal/2024-08/2024-08-12.md": "e57b03b929410f3111c894e43e1728ec", @@ -747,15 +749,27 @@ "_journal/2024-08/2024-08-15.md": "7c3a96a25643b62b0064bf32cb17d92f", "_journal/2024-08-17.md": "b06a551560c377f61a1b39286cd43cee", "_journal/2024-08/2024-08-16.md": "096d9147a9e3e7a947558f8dec763a2c", - "set/order.md": "49fcebf2e20a6f73571fea5ff09f0753", + "set/order.md": "66581eb2d882569b1591e660601caa55", "_journal/2024-08-18.md": "6f8aec69e00401b611db2a377a3aace5", "ontology/philosophy/properties.md": "41b32249d3e4c23d73ddb3a417d65a4c", - "_journal/2024-08-19.md": "82d3bfa01b4187a56a418f6e33bd10b3", + "_journal/2024-08-19.md": "94836e52ec04a72d3e1dbf3854208f65", "_journal/2024-08/2024-08-18.md": "6f8aec69e00401b611db2a377a3aace5", "_journal/2024-08/2024-08-17.md": "b06a551560c377f61a1b39286cd43cee", "calculus/bounds.md": "4add5fb7591087d0b3383c53dc62e365", "calculus/index.md": "5ee4d950533ae330ca5ef9e113fe87f3", - "x86-64/instructions/conditions.md": "c5571deac40ac2eeb8666f2d3b3c278e" + "x86-64/instructions/conditions.md": "c5571deac40ac2eeb8666f2d3b3c278e", + "_journal/2024-08-20.md": "e8bec308d1b29e411c6799ace7ef6571", + "algebra/arch-prop.md": "eccdd685f12898ed8679b558d19dc20a", + "_journal/2024-08/2024-08-19.md": "94836e52ec04a72d3e1dbf3854208f65", + "_journal/2024-08-21.md": "59e9483143ba6beec4f9ae2a09eb90a8", + "_journal/2024-08-22.md": "050235d5dc772b542773743b57ce3afe", + "_journal/2024-08/2024-08-21.md": "1637b8ec8475cf3eb4f41d1d86cbf5df", + "_journal/2024-08/2024-08-20.md": "e8bec308d1b29e411c6799ace7ef6571", + "_journal/2024-08-23.md": "3b2feab2cc927e267263cb1e9c173d50", + "set/natural-numbers.md": "97ca466daf1173ed8973db1d1a1935cc", + "_journal/2024-08-24.md": "9172485a4d1c47b5a181b96b68eb3ebc", + "_journal/2024-08/2024-08-23.md": "7b5a40e83d8f07ff54cd9708017d029c", + "_journal/2024-08/2024-08-22.md": "050235d5dc772b542773743b57ce3afe" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-08-24.md b/notes/_journal/2024-08-24.md new file mode 100644 index 0000000..5f6c410 --- /dev/null +++ b/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]]. \ No newline at end of file diff --git a/notes/_journal/2024-08-19.md b/notes/_journal/2024-08/2024-08-19.md similarity index 100% rename from notes/_journal/2024-08-19.md rename to notes/_journal/2024-08/2024-08-19.md diff --git a/notes/_journal/2024-08/2024-08-20.md b/notes/_journal/2024-08/2024-08-20.md new file mode 100644 index 0000000..c222cd6 --- /dev/null +++ b/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. \ No newline at end of file diff --git a/notes/_journal/2024-08/2024-08-21.md b/notes/_journal/2024-08/2024-08-21.md new file mode 100644 index 0000000..acefc7d --- /dev/null +++ b/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) \ No newline at end of file diff --git a/notes/_journal/2024-08/2024-08-22.md b/notes/_journal/2024-08/2024-08-22.md new file mode 100644 index 0000000..436fa93 --- /dev/null +++ b/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). \ No newline at end of file diff --git a/notes/_journal/2024-08/2024-08-23.md b/notes/_journal/2024-08/2024-08-23.md new file mode 100644 index 0000000..1934e93 --- /dev/null +++ b/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) \ No newline at end of file diff --git a/notes/algebra/arch-prop.md b/notes/algebra/arch-prop.md new file mode 100644 index 0000000..e88dbd5 --- /dev/null +++ b/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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). \ No newline at end of file diff --git a/notes/algebra/set.md b/notes/algebra/set.md index ca20be4..e70107e 100644 --- a/notes/algebra/set.md +++ b/notes/algebra/set.md @@ -1081,14 +1081,6 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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). - -END%% - %%ANKI Basic For sets $A$ and $B$, how is set $B^A$ pronounced? diff --git a/notes/c17/index.md b/notes/c17/index.md index 41a288f..00b7240 100644 --- a/notes/c17/index.md +++ b/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). END%% diff --git a/notes/c17/types.md b/notes/c17/types.md index a2f278c..09b06fd 100644 --- a/notes/c17/types.md +++ b/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). 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Enderton, *Elements of Set Theory* (New York: Academic Pre 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). + +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). + +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). + +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). diff --git a/notes/set/natural-numbers.md b/notes/set/natural-numbers.md new file mode 100644 index 0000000..0b72338 --- /dev/null +++ b/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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +END%% + +%%ANKI +Basic +What natural number corresponds to $\varnothing^{+++}$? +Back: $3$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What natural number corresponds to $\varnothing$? +Back: $0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +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). + +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). + +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). + +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). + +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). + +END%% + +## Bibliography + +* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). \ No newline at end of file diff --git a/notes/set/order.md b/notes/set/order.md index 2d4a59a..f97fbc9 100644 --- a/notes/set/order.md +++ b/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). 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). 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). 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). 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). 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). 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). 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). 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). 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). END%% diff --git a/notes/x86-64/registers.md b/notes/x86-64/registers.md index e0f0a21..61edc97 100644 --- a/notes/x86-64/registers.md +++ b/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. - + 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. - + 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. - + 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. - + END%% %%ANKI