diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index b03f2fa..fbd6e51 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -186,7 +186,14 @@ "infinity.png", "nan.png", "triangular-gnomon.png", - "pascals-triangle.png" + "pascals-triangle.png", + "function-bijective.png", + "function-injective.png", + "function-surjective.png", + "function-general.png", + "function-kernel.png", + "peano-system-i.png", + "peano-system-ii.png" ], "File Hashes": { "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8", @@ -326,7 +333,7 @@ "algebra/sequences/index.md": "208174a5a078b120fa11e296ad1d09c1", "_journal/2024-03-02.md": "08c3cae1df0079293b47e1e9556f1ce1", "_journal/2024-03/2024-03-01.md": "70da812300f284df72718dd32fc39322", - "algebra/sequences/triangular-numbers.md": "bf08ea7759b24defb7d5d3912cf04503", + "algebra/sequences/triangular-numbers.md": "aafaf54e5aff9ca3c7354591fce9f833", "algebra/sequences/square-numbers.md": "171f7c5a8dac088afba40923ab86c68e", "_journal/2024-03-03.md": "c4977a3778ed227b768c3f9ad5512670", "_journal/2024-03/2024-03-02.md": "8136792b0ee6e08232e4f60c88d461d2", @@ -578,7 +585,7 @@ "_journal/2024-06/2024-06-12.md": "f82dfa74d0def8c3179d3d076f94558e", "_journal/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307", "_journal/2024-06/2024-06-13.md": "e2722a00585d94794a089e8035e05728", - "set/functions.md": "6716f8a32af73e5a4d1b2cbf6987b60f", + "set/functions.md": "59e449d6756b57c846cdf07b0a1b4330", "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4", "_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307", "lambda-calculus/beta-reduction.md": "a8e2825c84e842ceef7aa638a493b91a", @@ -738,7 +745,7 @@ "_journal/2024-08/2024-08-15.md": "7c3a96a25643b62b0064bf32cb17d92f", "_journal/2024-08-17.md": "b06a551560c377f61a1b39286cd43cee", "_journal/2024-08/2024-08-16.md": "da1127a1985074a3930b4c3512344025", - "set/order.md": "3bf63dd9c8ce6d2b4c6905dab0bd4aad", + "set/order.md": "b69f922200514975b7a7028eef030b59", "_journal/2024-08-18.md": "6f8aec69e00401b611db2a377a3aace5", "ontology/philosophy/properties.md": "41b32249d3e4c23d73ddb3a417d65a4c", "_journal/2024-08-19.md": "94836e52ec04a72d3e1dbf3854208f65", @@ -755,7 +762,7 @@ "_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": "f37647a51f457cb7d335e4e4fff227de", + "set/natural-numbers.md": "bf73972ec1ca619ba0124169c25b1c39", "_journal/2024-08-24.md": "563fad24740e44734a87d7c3ec46cec4", "_journal/2024-08/2024-08-23.md": "7b5a40e83d8f07ff54cd9708017d029c", "_journal/2024-08/2024-08-22.md": "050235d5dc772b542773743b57ce3afe", @@ -797,7 +804,12 @@ "_journal/2024-09/2024-09-10.md": "71a766783213f58552990b3ab1baeb50", "_journal/2024-09/2024-09-08.md": "0949eaf8df8d7e35cc0734d3a823921a", "_journal/2024-09/2024-09-07.md": "807e46a75e8b4b414141fb0c7d3f03e4", - "_journal/2024-09/2024-09-06.md": "7ea6a87f77cf49943eb76dd1052bd736" + "_journal/2024-09/2024-09-06.md": "7ea6a87f77cf49943eb76dd1052bd736", + "_journal/2024-09-14.md": "774019f651e728faa288041ce4b265d3", + "_journal/2024-09/2024-09-13.md": "8c8f33fdd8242e5ab9adaa797dea7995", + "_journal/2024-09/2024-09-12.md": "30968fa3d73c005bdb4acc2025b34e11", + "_journal/2024-09-15.md": "a203f489d0205246b9b625354123046c", + "_journal/2024-09/2024-09-14.md": "1050e9ae0dfe4196a419105c43c2162f" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-09-15.md b/notes/_journal/2024-09-15.md new file mode 100644 index 0000000..32f6825 --- /dev/null +++ b/notes/_journal/2024-09-15.md @@ -0,0 +1,9 @@ +--- +title: "2024-09-15" +--- + +- [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/2024-09-12.md b/notes/_journal/2024-09/2024-09-12.md similarity index 61% rename from notes/_journal/2024-09-12.md rename to notes/_journal/2024-09/2024-09-12.md index 1be126e..2112cbc 100644 --- a/notes/_journal/2024-09-12.md +++ b/notes/_journal/2024-09/2024-09-12.md @@ -2,7 +2,7 @@ title: "2024-09-12" --- -- [ ] Anki Flashcards +- [x] Anki Flashcards - [x] KoL - [x] OGS - [ ] Sheet Music (10 min.) diff --git a/notes/_journal/2024-09/2024-09-13.md b/notes/_journal/2024-09/2024-09-13.md new file mode 100644 index 0000000..c1c470c --- /dev/null +++ b/notes/_journal/2024-09/2024-09-13.md @@ -0,0 +1,11 @@ +--- +title: "2024-09-13" +--- + +- [x] Anki Flashcards +- [x] KoL +- [ ] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Watched [Query Execution 1](https://www.youtube.com/watch?v=I0QdCSu06_o&list=PLSE8ODhjZXjaKScG3l0nuOiDTTqpfnWFf&index=13) and [Parallel Query Execution](https://www.youtube.com/watch?v=FG_wr-0QYg4&list=PLSE8ODhjZXjaKScG3l0nuOiDTTqpfnWFf&index=14). \ No newline at end of file diff --git a/notes/_journal/2024-09/2024-09-14.md b/notes/_journal/2024-09/2024-09-14.md new file mode 100644 index 0000000..b6d1622 --- /dev/null +++ b/notes/_journal/2024-09/2024-09-14.md @@ -0,0 +1,11 @@ +--- +title: "2024-09-14" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Notes on [[natural-numbers#Peano System|Peano systems]]. \ No newline at end of file diff --git a/notes/algebra/sequences/triangular-numbers.md b/notes/algebra/sequences/triangular-numbers.md index 5c6ef91..b65cbcc 100644 --- a/notes/algebra/sequences/triangular-numbers.md +++ b/notes/algebra/sequences/triangular-numbers.md @@ -14,7 +14,7 @@ The $n$th term of the **triangular numbers** $(T_n)_{n \geq 0}$ is the sum of wh %%ANKI Basic What is a polygonal number? -Back: A number of pebbles that can be arranged into the shape of a regular polygon. +Back: A number of pebbles that can be arranged into the shape of a regular filled polygon. Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% diff --git a/notes/set/functions.md b/notes/set/functions.md index c4bff1f..5bcdd04 100644 --- a/notes/set/functions.md +++ b/notes/set/functions.md @@ -1641,6 +1641,73 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +### Closures + +If $S$ is a function and $A$ is a subset of $\mathop{\text{dom}}S$, then $A$ is said to be **closed** under $S$ if and only if whenever $x \in A$, then $S(x) \in A$. This is equivalently expressed as $S[\![A]\!] \subseteq A$. + +%%ANKI +Basic +Let $A$ be closed under $S$. Then $A$ is a subset of what other set? +Back: $\mathop{\text{dom}}S$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $A$ be closed under $S$. With maximum specificity, what kind of mathematical object is $A$? +Back: A set. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $A$ be closed under $S$. With maximum specificity, what kind of mathematical object is $S$? +Back: A function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +In FOL, what does it mean for set $A$ to be closed under function $S$? +Back: $\forall x \in A, S(x) \in A$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What concept is being expressed in "$\forall x \in A, S(x) \in A$"? +Back: Set $A$ is closed under $S$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How can we more compactly express "$\forall x \in A, S(x) \in A$"? +Back: $S[\![A]\!] \subseteq A$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +If $S[\![A]\!] \subseteq A$, then {1:$A$} is closed {2:under} {1:$S$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Suppose $A$ is closed under function $S$. What imagery does the term "closed" invoke? +Back: Applying a member of $A$ to $S$ always yields an element in $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ## Kernels Let $F \colon A \rightarrow B$. Define [[relations#Equivalence Relations|equivalence relation]] $\sim$ as $$x \sim y \Leftrightarrow f(x) = f(y)$$ diff --git a/notes/set/images/peano-system-i.png b/notes/set/images/peano-system-i.png new file mode 100644 index 0000000..dc1ee99 Binary files /dev/null and b/notes/set/images/peano-system-i.png differ diff --git a/notes/set/images/peano-system-ii.png b/notes/set/images/peano-system-ii.png new file mode 100644 index 0000000..50f3c13 Binary files /dev/null and b/notes/set/images/peano-system-ii.png differ diff --git a/notes/set/natural-numbers.md b/notes/set/natural-numbers.md index 61ee1fe..3111ec9 100644 --- a/notes/set/natural-numbers.md +++ b/notes/set/natural-numbers.md @@ -369,6 +369,175 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +## Peano System + +A **Peano system** is a triple $\langle N, S, e \rangle$ consisting of a set $N$, a function $S \colon N \rightarrow N$, and a member $e \in N$ such that the following three conditions are met: + +* $e \not\in \mathop{\text{ran}}{S}$; +* $S$ is one-to-one; +* Any subset $A$ of $N$ that contains $e$ and is closed under $S$ equals $N$ itself. + +%%ANKI +Basic +A Peano system is a tuple consisting of how many members? +Back: $3$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $N$? +Back: A set. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $S$? +Back: A function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider Peano system $\langle N, S, e \rangle$. What is the domain of $S$? +Back: $N$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider Peano system $\langle N, S, e \rangle$. What is the codomain of $S$? +Back: $N$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, 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). + +END%% + +%%ANKI +Basic +In Peano system $\langle N, S, e \rangle$, $e$ is a member of what set? +Back: $N$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +In Peano system $\langle N, S, e \rangle$, $e$ is explicitly *not* a member of what set? +Back: $\mathop{\text{ran}}S$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +Consider Peano system $\langle N, S, e \rangle$. Then {1:$e$} $\not\in$ {1:$\mathop{\text{ran} }S$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider Peano system $\langle N, S, e \rangle$. Function $S$ satisfies what additional condition? +Back: $S$ is one-to-one. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider Peano system $\langle N, S, e \rangle$. What two conditions must be satisfied for $A \subseteq N$ to coincide with $N$? +Back: $e \in A$ and $A$ is closed under $S$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What condition of Peano system $\langle N, S, e \rangle$ generalizes the induction principle of $\omega$? +Back: Any set $A \subseteq N$ containing $e$ and closed under $S$ coincides with $N$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What name is given to the condition of Peano systems involving closures? +Back: The Peano induction postulate. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +The Peano induction postulate of $\langle N, S, e \rangle$ implies $N$ is the smallest set satisfying what? +Back: That contains $e$ and is closed under $S$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $\langle N, S, e \rangle$ be a Peano system. *Why* can't there be an $A \subset N$ containing $e$ and closed under $S$? +Back: The Peano induction postulate states $A$ *must* coincide with $N$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +*Why* does Peano system $\langle N, S, e \rangle$ have condition $e \not\in \mathop{\text{ran}}S$? +Back: To avoid cycles in repeated applications of $S$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Which condition of Peano system $\langle N, S, e \rangle$ does the following depict? +![[peano-system-i.png]] +Back: $e \not\in \mathop{\text{ran}}S$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +*Why* does Peano system $\langle N, S, e \rangle$ have condition "$S$ is one-to-one"? +Back: To avoid two members of $N$ mapping to the same element. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Which condition of Peano system $\langle N, S, e \rangle$ does the following depict? +![[peano-system-ii.png]] +Back: $S$ is one-to-one. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the Peano induction postulate? +Back: Given Peano system $\langle N, S, e \rangle$, a set $A \subseteq N$ containing $e$ and closed under $S$ coincides with $N$. +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 2e7478b..40720c1 100644 --- a/notes/set/order.md +++ b/notes/set/order.md @@ -487,7 +487,7 @@ END%% %%ANKI Basic -Consider an equivalence class of $x$ (modulo $R$). What kind of mathematical object is $R$? +Consider an equivalence class of $x$ (modulo $R$). With maximum specificity, what kind of mathematical object is $R$? Back: A relation. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).