From 2bf0ac9adbc5977730c45be53808c3c60e9654bc Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Thu, 19 Sep 2024 20:04:19 -0600 Subject: [PATCH] Notes on transitive sets. --- .../plugins/obsidian-to-anki-plugin/data.json | 14 +- notes/_journal/2024-09-19.md | 4 +- notes/set/natural-numbers.md | 130 ++++++++++++++++++ notes/set/relations.md | 8 -- 4 files changed, 143 insertions(+), 13 deletions(-) diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 27203f6..a634e1d 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -193,7 +193,8 @@ "function-general.png", "function-kernel.png", "peano-system-i.png", - "peano-system-ii.png" + "peano-system-ii.png", + "relation-ordering-example.png" ], "File Hashes": { "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8", @@ -559,7 +560,7 @@ "_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": "9323fc61ee983487e16f6dd4c1629957", + "set/relations.md": "18d3f9a6bec2a23a576910750fd2158d", "_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2", "_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381", "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8", @@ -762,7 +763,7 @@ "_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": "1135fcfce9759ed4c5689df59b0c9fe3", + "set/natural-numbers.md": "353d1eef7b50fa8f635adc928df734fa", "_journal/2024-08-24.md": "563fad24740e44734a87d7c3ec46cec4", "_journal/2024-08/2024-08-23.md": "7b5a40e83d8f07ff54cd9708017d029c", "_journal/2024-08/2024-08-22.md": "050235d5dc772b542773743b57ce3afe", @@ -810,7 +811,12 @@ "_journal/2024-09/2024-09-12.md": "30968fa3d73c005bdb4acc2025b34e11", "_journal/2024-09-15.md": "0e5d1ecd73edf343d3a268b25140a921", "_journal/2024-09/2024-09-14.md": "1050e9ae0dfe4196a419105c43c2162f", - "_journal/2024-09-16.md": "bd3f811538b8e8b288b3376246471c0a" + "_journal/2024-09-16.md": "bd3f811538b8e8b288b3376246471c0a", + "_journal/2024-09/2024-09-18.md": "5cba3e6646783019db268af3b353c5d8", + "_journal/2024-09/2024-09-17.md": "caea0dab26b0970c045ccd9e5f2f3765", + "_journal/2024-09/2024-09-16.md": "7dd800d514051dd36c33c14623c7c5c8", + "_journal/2024-09/2024-09-15.md": "0e5d1ecd73edf343d3a268b25140a921", + "_journal/2024-09-19.md": "612f08e8f9ce35f71378e2c4a636c862" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-09-19.md b/notes/_journal/2024-09-19.md index 1ad2e05..ebba896 100644 --- a/notes/_journal/2024-09-19.md +++ b/notes/_journal/2024-09-19.md @@ -6,4 +6,6 @@ title: "2024-09-19" - [x] KoL - [x] OGS - [ ] Sheet Music (10 min.) -- [ ] Korean (Read 1 Story) \ No newline at end of file +- [ ] Korean (Read 1 Story) + +* Notes on [[natural-numbers#Transitivity|transitive sets]]. \ No newline at end of file diff --git a/notes/set/natural-numbers.md b/notes/set/natural-numbers.md index d9cbdcb..5906de3 100644 --- a/notes/set/natural-numbers.md +++ b/notes/set/natural-numbers.md @@ -538,6 +538,136 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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: + +* $x \in a \in A \Rightarrow x \in A$ +* $\bigcup A \subseteq A$ +* $a \in A \Rightarrow a \subseteq A$ +* $A \subseteq \mathscr{P}A$ + +%%ANKI +Basic +What does it mean for $A$ to be a transitive set? +Back: Every member of a member of $A$ is itself a member of $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +In what way is the term "transitive set" ambiguous? +Back: This term can also be used to describe a transitive relation. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +A transitive {1:set} is to {2:membership} whereas a transitive {2:relation} is to {1:related}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +$A$ is a transitive set iff {$x \in a \in A$} $\Rightarrow$ {$x \in A$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +$A$ is a transitive set iff {$\bigcup A$} $\subseteq$ {$A$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +$A$ is a transitive set iff {$a \in A$} $\Rightarrow$ {$a \subseteq A$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +$A$ is a transitive set iff {$A$} $\subseteq$ {$\mathscr{P} A$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Is $\varnothing$ a transitive set? +Back: Yes. + +END%% + +%%ANKI +Basic +*Why* isn't $\{0, 1\}$ a transitive set? +Back: N/A. It is. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +*Why* isn't $\{1\}$ a transitive set? +Back: Because $0 \in 1$ but $0 \not\in \{1\}$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +*Why* isn't $\{\varnothing\}$ a transitive set? +Back: N/A. It is. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +*Why* isn't $\{\{\varnothing\}\}$ a transitive set? +Back: Because $\varnothing \in \{\varnothing\}$ but $\varnothing \not\in \{\{\varnothing\}\}$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Suppose $a$ is a transitive set. *Why* does $\bigcup a \cup a = a$? +Back: Because transitivity holds if and only if $\bigcup a \subseteq a$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Suppose $A \cup B = A$. What relation immediately follows? +Back: $B \subseteq A$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Suppose $A \cap B = A$. What relation immediately follows? +Back: $B = A$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +$A$ is a transitive set iff {$\bigcup$}$A^+ =$ {$A$}. +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/relations.md b/notes/set/relations.md index 4da550c..266a17b 100644 --- a/notes/set/relations.md +++ b/notes/set/relations.md @@ -1105,14 +1105,6 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% -%%ANKI -Basic -The term "transitive" is used to describe what kind of mathematical object? -Back: Relations. -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - ## Connected A binary relation $R$ on set $A$ is said to be **connected** if for any *distinct* $x, y \in A$, either $xRy$ or $yRx$. The relation is **strongly connected** if for *all* $x, y \in A$, either $xRy$ or $yRx$.