From 4081f0facb14e47a1cb80bfe893e4c7c3f201048 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Sat, 17 Aug 2024 12:05:51 -0600 Subject: [PATCH] Strict preorders/partial orders. --- .../plugins/obsidian-to-anki-plugin/data.json | 8 +- notes/_journal/2024-08-17.md | 11 ++ notes/_journal/{ => 2024-08}/2024-08-16.md | 0 notes/hashing/index.md | 2 +- notes/set/relations.md | 155 +++++++++++++++++- 5 files changed, 166 insertions(+), 10 deletions(-) create mode 100644 notes/_journal/2024-08-17.md rename notes/_journal/{ => 2024-08}/2024-08-16.md (100%) diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 82cf373..2190a91 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -486,7 +486,7 @@ "_journal/2024-05/2024-05-16.md": "9fdfadc3f9ea6a4418fd0e7066d6b10c", "_journal/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c", "hashing/direct-addressing.md": "f75cc22e74ae974fe4f568a2ee9f951f", - "hashing/index.md": "3a0fb5369b0d3adac5ac13b8b3689e2e", + "hashing/index.md": "f1dbb16667b056f2ea00db8db805c54f", "set/classes.md": "6776b4dc415021e0ef60b323b5c2d436", "_journal/2024-05-19.md": "fddd90fae08fab9bd83b0ef5d362c93a", "_journal/2024-05/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c", @@ -534,7 +534,7 @@ "_journal/2024-06/2024-06-04.md": "52b28035b9c91c9b14cef1154c1a0fa1", "_journal/2024-06-06.md": "3f9109925dea304e7172df39922cc95a", "_journal/2024-06/2024-06-05.md": "b06a0fa567bd81e3b593f7e1838f9de1", - "set/relations.md": "feaa8115db02fbbda77b483c37f183ca", + "set/relations.md": "60fed0d4642d214767c077ca44983357", "_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2", "_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381", "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8", @@ -717,7 +717,9 @@ "_journal/2024-08-15.md": "fabf6e09bfd99cd180a4c674f83ebcb9", "_journal/2024-08/2024-08-14.md": "f7d1dede5ab6e4634ad9de3d3426c6f7", "_journal/2024-08-16.md": "a25c680684bcffc6a38cebbb448d9d97", - "_journal/2024-08/2024-08-15.md": "7c3a96a25643b62b0064bf32cb17d92f" + "_journal/2024-08/2024-08-15.md": "7c3a96a25643b62b0064bf32cb17d92f", + "_journal/2024-08-17.md": "f9305a6a34de9a510260bed7d2bb1aca", + "_journal/2024-08/2024-08-16.md": "096d9147a9e3e7a947558f8dec763a2c" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-08-17.md b/notes/_journal/2024-08-17.md new file mode 100644 index 0000000..656b2ef --- /dev/null +++ b/notes/_journal/2024-08-17.md @@ -0,0 +1,11 @@ +--- +title: "2024-08-17" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Notes on strict preorders/partial orders. \ No newline at end of file diff --git a/notes/_journal/2024-08-16.md b/notes/_journal/2024-08/2024-08-16.md similarity index 100% rename from notes/_journal/2024-08-16.md rename to notes/_journal/2024-08/2024-08-16.md diff --git a/notes/hashing/index.md b/notes/hashing/index.md index e404326..502d991 100644 --- a/notes/hashing/index.md +++ b/notes/hashing/index.md @@ -703,7 +703,7 @@ END%% %%ANKI Basic What is it that universal hashing makes impossible? -Back: The ability of an adversary forcing the worst-case running time of hash table operations. +Back: Forcing the worst-case running time of hash table operations. Tags: hashing::random hashing::universal END%% diff --git a/notes/set/relations.md b/notes/set/relations.md index e765582..ad4b12e 100644 --- a/notes/set/relations.md +++ b/notes/set/relations.md @@ -1358,7 +1358,7 @@ $R$ is a **preorder on $A$** iff $R$ is a binary relation that is reflexive on s %%ANKI Basic What is a preorder on $A$? -Back: A binary relation reflexive on $A$ and transitive. +Back: A binary relation on $A$ that is reflexive on $A$ and transitive. 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%% @@ -1411,11 +1411,82 @@ Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia. END%% +A **strict preorder** replaces reflexivity with irreflexivity. That is, $R$ is a strict preorder on $A$ iff $R$ is a binary relation on set $A$ that is irreflexive on $A$ and transitive. + +%%ANKI +Basic +What distinguishes a preorder from a strict preorder? +Back: Strict preorders are irreflexive. +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%% + +%%ANKI +Basic +What is a strict preorder on $A$? +Back: A binary relation on $A$ that is irreflexive on $A$ and transitive. +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%% + +%%ANKI +Basic +What makes a strict preorder more strict than a non-strict preorder? +Back: Strict preorders do not allow relating members to themselves. +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%% + +%%ANKI +Basic +*Why* isn't $R = \{\langle a, a \rangle\}$ a strict preorder on $\{a\}$? +Back: $R$ isn't irreflexive. +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%% + +%%ANKI +Basic +*Why* isn't $R = \{\langle a, b \rangle, \langle b, c \rangle, \langle a, c \rangle\}$ a strict preorder on $\{a, b, c\}$? +Back: N/A. It is. +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%% + +%%ANKI +Basic +*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle \}$ a strict preorder on $\{a, b\}$? +Back: $R$ isn't irreflexive. +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%% + +%%ANKI +Cloze +A {1:strict} preorder is equivalent to a {1:strict} partial order. +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%% + +%%ANKI +Basic +*Why* is a strict preorder also a strict partial order? +Back: Irreflexivity and transitivity imply asymmetry (and antisymmetry). +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%% + +%%ANKI +Basic +What equivalence in order theory serves as a mnemonic for "irreflexivity and transitivity imply asymmetry"? +Back: A strict preorder is equivalent to a strict partial order. +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%% + ## Partial Orders -$R$ is a **partial order on $A$** iff $R$ is a binary relation on set $A$ that is reflexive on $A$, antisymmetric, and transitive. - -In other words, a partial order is an antisymmetric preorder. +$R$ is a **partial order on $A$** iff $R$ is a binary relation on set $A$ that is reflexive on $A$, antisymmetric, and transitive. In other words, a partial order is an antisymmetric preorder. %%ANKI Basic @@ -1506,7 +1577,7 @@ END%% %%ANKI Basic -*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ a partial order on $\{a, b\}$? +*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ a partial order on $\{a, b, c\}$? Back: N/A. It is. 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). @@ -1514,12 +1585,84 @@ END%% %%ANKI Basic -*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle, \langle c, b \rangle\}$ a partial order on $\{a, b\}$? +*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. 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%% +A **strict partial order** replaces reflexivity with irreflexivity. That is, $R$ is a strict partial order on $A$ iff $R$ is a binary relation on set $A$ that is irreflexive on $A$, antisymmetric, and transitive. + +%%ANKI +Basic +What distinguishes a partial order from a strict partial order? +Back: Strict partial orders are irreflexive. +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 +Basic +What is a strict partial order on $A$? +Back: A binary relation on $A$ that is irreflexive on $A$, antisymmetric, and 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%% + +%%ANKI +Basic +What makes a strict partial order more strict than a non-strict partial order? +Back: Strict partial orders do not allow relating members to themselves. +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 {$<$} typically denote 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. +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 +Basic +*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle\}$ a strict partial order? +Back: N/A. The question must provide a reference set. +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 +Basic +*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ a strict partial order on $\{a, b, c\}$? +Back: Because it isn't irreflexive. +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 +Basic +*Why* isn't $R = \{\langle a, c \rangle, \langle b, c \rangle\}$ a strict partial order on $\{a, b, c\}$? +Back: N/A. It is. +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 +Basic +*Why* isn't $R = \{\langle a, b \rangle, \langle b, c \rangle, \langle c, b \rangle\}$ a strict partial order on $\{a, b\}$? +Back: It is neither antisymmetric nor 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%% + ## Equivalence Relations $R$ is an **equivalence relation on $A$** iff $R$ is a binary relation on set $A$ that is reflexive on $A$, symmetric, and transitive.