From 2e77d20f15163c04c761890f0ef4c70270877273 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Sun, 14 Jul 2024 09:03:36 -0600 Subject: [PATCH] Additional notes on equivalence relations. --- .../plugins/obsidian-to-anki-plugin/data.json | 4 +- notes/_journal/2024-07-14.md | 3 +- notes/set/relations.md | 195 ++++++++++++++++++ 3 files changed, 199 insertions(+), 3 deletions(-) diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 6ff9cd8..8c140d5 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -509,7 +509,7 @@ "_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": "63cdf623940e4076d127440574f6b6a2", + "set/relations.md": "b98d316d6e7595c5406352541cc7f9ee", "_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2", "_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381", "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8", @@ -601,7 +601,7 @@ "ontology/index.md": "c523aa6652b285a0f1c053cb77be6f85", "ontology/permissivism.md": "5b66dd065aa66d5a2624eda032d75b94", "ontology/properties.md": "d417db0cecf11b1ed2e17f165d879fa5", - "_journal/2024-07-14.md": "d790f7f6b13285b91cd972d927c20f96", + "_journal/2024-07-14.md": "3e04720086dca16f1c6b7412ec1706fd", "_journal/2024-07/2024-07-13.md": "60e8eb09812660a2f2bf86ffafab5714" }, "fields_dict": { diff --git a/notes/_journal/2024-07-14.md b/notes/_journal/2024-07-14.md index 5775c98..e24107a 100644 --- a/notes/_journal/2024-07-14.md +++ b/notes/_journal/2024-07-14.md @@ -8,4 +8,5 @@ title: "2024-07-14" - [ ] Sheet Music (10 min.) - [ ] Korean (Read 1 Story) -* Notes on infinite Cartesian products and their relation to the axiom of choice. \ No newline at end of file +* Notes on [[set#Cartesian Product|infinite Cartesian products]] and their relation to the [[set/index#Infinite Cartesian Product Form|axiom of choice]]. +* Initial notes on [[relations#Equivalence Relations|equivalence relations]]. \ No newline at end of file diff --git a/notes/set/relations.md b/notes/set/relations.md index fc3dc37..c411107 100644 --- a/notes/set/relations.md +++ b/notes/set/relations.md @@ -601,6 +601,201 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +## Equivalence Relations + +Given relation $R$ and set $A$, $R$ is an **equivalence relation on $A$** iff $R$ is a binary relation on $A$ that is reflexive on $A$, symmetric, and transitive: + +* $R$ is **reflexive on $A$** if $xRx$ for all $x \in A$. +* $R$ is **symmetric** if whenever $xRy$, then $yRx$. +* $R$ is **transitive** if whenever $xRy$ and $yRz$, then $xRz$. + +%%ANKI +Cloze +Binary relation $R$ is {reflexive on $A$} iff {$xRx$ for all $x \in A$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Why is it incorrect to ask if $R$ is reflexive? +Back: We have to ask if $R$ is reflexive on some reference set. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive? +Back: N/A. The question must provide a reference set. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $a$? +Back: N/A. We must ask if $R$ is reflexive on a set. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $\{a\}$? +Back: Yes. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $\{a, b\}$? +Back: No. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, *why* isn't $R$ reflexive on $\{a, b\}$? +Back: Because $\langle b, b \rangle \not\in R$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +If $xRx$ for all $x \in A$, $R$ is said to be reflexive {on} $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +Binary relation $R$ is {symmetric} iff {$xRy \Rightarrow yRx$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, is $R$ symmetric? +Back: No. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, what additional member(s) must be added to make $R$ symmetric? +Back: $\langle b, a \rangle$ and $\langle c, b \rangle$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, a \rangle, \langle b, b \rangle\}$, which of reflexivity (on $\{a, b\}$), symmetry, and transitivity does $R$ exhibit? +Back: Reflexivity on $\{a, b\}$ and symmetry. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +Binary relation $R$ is {transitive} iff {$xRy \land yRz \Rightarrow xRz$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, is $R$ transitive? +Back: No. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, what additional member(s) must be added to make $R$ transitive? +Back: Just $\langle a, c \rangle$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is an equivalence relation on $A$? +Back: A binary relation on $A$ that is reflexive on $A$, symmetric, and transitive. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +An equivalence relation on $A$ is a {$2$}-ary relation on $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, a \rangle\}$, is $R$ an equivalence relation? +Back: N/A. The question must provide a reference set. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, a \rangle\}$, is $R$ an equivalence relation on $\{a\}$? +Back: Yes. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ an equivalence relation on $\{a\}$? +Back: No. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ an equivalence relation on $\{a\}$? +Back: $R$ is neither symmetric nor transitive. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Which of symmetric relations and transitive relations is more general? +Back: N/A. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Which of equivalence relations on $A$ and symmetric relations is more general? +Back: Symmetric relations. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Which of binary relations on $A$ and equivalence relations on $A$ is more general? +Back: Binary relations on $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ## Bibliography * “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).