diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 6a53976..3360df7 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -322,7 +322,7 @@ "_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f", "_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b", "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53", - "set/index.md": "c366c641119b2f83558666eede68eef0", + "set/index.md": "fb066e22d2531f530e2899df975e4e35", "set/graphs.md": "75f0ee994436ae39f7ba94a4eb73435a", "_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb", "_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391", @@ -508,7 +508,7 @@ "_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": "3355df45182a017a56670594ba8d5a13", + "set/relations.md": "83e38548017dda4fa6371fa1b312b2e2", "_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2", "_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381", "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8", @@ -534,7 +534,7 @@ "_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": "b436a00b2d21aa6870b8c5f0684d0e51", + "set/functions.md": "8d2f0ef04e32de2de5054127f6970f18", "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4", "_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307", "lambda-calculus/beta-reduction.md": "bd7ed2d1b8aae2e584c3e7be1d116170", @@ -568,7 +568,7 @@ "_journal/2024-06-28.md": "327d1abef4be0d1526a585dcfe03db5a", "_journal/2024-06/2024-06-27.md": "237c73268a28f652985a5ef7ca7e188e", "_journal/2024-06/2024-06-26.md": "9c5d7e6395496736f2f268e9fdba117f", - "_journal/2024-06-29.md": "d4bdb04616774ff741886e94e5f2e385", + "_journal/2024-06-29.md": "9d43f4f33e03a48aa08e13bb5be365e0", "_journal/2024-06/2024-06-28.md": "3f6a47a6324918b6c3af6b9549663372" }, "fields_dict": { diff --git a/notes/_journal/2024-06-29.md b/notes/_journal/2024-06-29.md index b2666f8..8dec021 100644 --- a/notes/_journal/2024-06-29.md +++ b/notes/_journal/2024-06-29.md @@ -2,8 +2,10 @@ title: "2024-06-29" --- -- [ ] Anki Flashcards +- [x] Anki Flashcards - [x] KoL - [x] OGS - [ ] Sheet Music (10 min.) -- [ ] Korean (Read 1 Story) \ No newline at end of file +- [ ] Korean (Read 1 Story) + +* Left and right inverses. Touched on the Axiom of Choice. \ No newline at end of file diff --git a/notes/set/functions.md b/notes/set/functions.md index 92c6e0b..c62028f 100644 --- a/notes/set/functions.md +++ b/notes/set/functions.md @@ -9,7 +9,7 @@ tags: ## Overview -A **function** $F$ is a single-valued [[relations|relation]]. We say $F$ **maps $A$ into $B$**, denoted $F \colon A \rightarrow B$, if and only if $F$ is a function, $\mathop{\text{dom}}A$, and $\mathop{\text{ran}}F \subseteq B$. +A **function** $F$ is a single-valued [[relations|relation]]. We say $F$ **maps $A$ into $B$**, denoted $F \colon A \rightarrow B$, if and only if $F$ is a function, $\mathop{\text{dom}}F = A$, and $\mathop{\text{ran}}F \subseteq B$. %%ANKI Basic @@ -122,6 +122,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +Is $\varnothing$ a function? +Back: Yes. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ## Injections A function is **injective** or **one-to-one** if each element of the codomain is mapped to by at most one element of the domain. @@ -258,6 +266,129 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +### Left Inverses + +Assume that $F \colon A \rightarrow B$ is a function and $A \neq \varnothing$. Then there exists a function $G \colon B \rightarrow A$ (a **left inverse**) such that $G \circ F = I_A$ if and only if $F$ is one-to-one. + +%%ANKI +Basic +What is the most specific mathematical object that describes a left inverse? +Back: A function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is a left inverse of $F \colon A \rightarrow B$ defined? +Back: As a function $G \colon B \rightarrow A$ such that $G \circ F = I_A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is a left inverse of set $A$ defined? +Back: N/A. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider $F \colon A \rightarrow B$. If $F$ has a left inverse, what is its domain? +Back: $B$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What does $I_A$ usually denote? +Back: The identity function on set $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is the identity function on set $B$ denoted? +Back: $I_B$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider $F \colon A \rightarrow B$. If $F$ has a left inverse, what is its codomain? +Back: $A$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $G$ be a left inverse of $F \colon A \rightarrow B$. How can we more simply write $G \circ F$? +Back: $I_A$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $G$ be a left inverse of $F \colon A \rightarrow B$. How can we more simply write $F \circ G$? +Back: N/A. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $F$ be a left inverse of function $G$. How do they interestingly compose? +Back: As $F \circ G$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +Suppose $F \colon A \rightarrow B$ and {1:$A \neq \varnothing$}. $F$ has a {2:left} inverse iff $F$ is {3:one-to-one}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Does proving "left inverses iff injective" rely on AoC? +Back: No. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What are the hypotheses of "left inverses iff injective"? +Back: Suppose $F \colon A \rightarrow B$ such that $A \neq \varnothing$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $F \colon A \rightarrow B$. *Why* does "left inverses iff injective" require $A \neq \varnothing$? +Back: Because a mapping from $B$ to $\varnothing$ cannot be a function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $F \colon A \rightarrow B$ and $A \neq \varnothing$. *Why* does "left inverses iff injective" require AoC? +Back: It doesn't. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ## Surjections A function is **surjective** or **onto** if each element of the codomain is mapped to by at least one element of the domain. That is, **$F$ maps $A$ onto $B$** if and only if $F$ is a function, $\mathop{\text{dom}}A$, and $\mathop{\text{ran}}F = B$. @@ -369,6 +500,120 @@ Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 202 END%% +### Right Inverses + +Assume that $F \colon A \rightarrow B$ is a function and $A \neq \varnothing$. Then there exists a function $G \colon B \rightarrow A$ (a right inverse) such that $F \circ G = I_B$ if and only if $F$ maps $A$ onto $B$. + +%%ANKI +Basic +What is the most specific mathematical object that describes a right inverse? +Back: A function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is a right inverse of $F \colon A \rightarrow B$ defined? +Back: As a function $G \colon B \rightarrow A$ such that $F \circ G = I_B$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is a right inverse of set $A$ defined? +Back: N/A. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +{1:Left} inverses are to {2:injections} whereas {2:right} inverses are to {1:surjections}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider $F \colon A \rightarrow B$. If $F$ has a right inverse, what is its domain? +Back: $B$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider $F \colon A \rightarrow B$. If $F$ has a right inverse, what is its codomain? +Back: $A$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $G$ be a right inverse of $F \colon A \rightarrow B$. How can we more simply write $G \circ F$? +Back: N/A. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $G$ be a right inverse of $F \colon A \rightarrow B$. How can we more simply write $F \circ G$? +Back: The identity function on $B$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $F$ be a right inverse of function $G$. How do they interestingly compose? +Back: As $G \circ F$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +Suppose $F \colon A \rightarrow B$ and {1:$A \neq \varnothing$}. $F$ has a {2:right} inverse iff $F$ is {3:onto $B$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Does proving "right inverses iff surjective" rely on AoC? +Back: Yes. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What are the hypotheses of "right inverses iff surjective"? +Back: Suppose $F \colon A \rightarrow B$ such that $A \neq \varnothing$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $F \colon A \rightarrow B$. *Why* does "right inverses iff surjective" require $A \neq \varnothing$? +Back: Because a mapping from $B$ to $\varnothing$ cannot be a function. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $F \colon A \rightarrow B$ and $A \neq \varnothing$. *Why* does "right inverses iff surjective" require AoC? +Back: There is no other mechanism for choosing an $x \in A$ for *each* $y \in B$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ## Bijections A function is **bijective** or a **one-to-one correspondence** if each element of the codomain is mapped to by exactly one element of the domain. @@ -441,6 +686,14 @@ END%% ## Inverses Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}$$ +%%ANKI +Basic +What is the most specific mathematical object that describes an inverse? +Back: A relation. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + %%ANKI Basic What kind of mathematical object does the inverse operation apply to? @@ -656,6 +909,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +Consider function $F \colon \varnothing \rightarrow B$. What is $F^{-1}$? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ## Compositions Let $F$ and $G$ be arbitrary sets. The **composition** of $F$ and $G$ is the set $$F \circ G = \{\langle u, v \rangle \mid \exists t, uGt \land tFv \}$$ @@ -997,14 +1258,6 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% -%%ANKI -Basic -How is $F^{-1}[\![A]\!]$ defined using set-builder notation? -Back: $F^{-1}[\![A]\!] = \{x \in \mathop{\text{dom}} F \mid F(x) \in A \}$ -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - %%ANKI Basic Let $F$ be an arbitrary set. What is $F[\![\varnothing]\!]$? diff --git a/notes/set/index.md b/notes/set/index.md index d3c8f68..2a3f75d 100644 --- a/notes/set/index.md +++ b/notes/set/index.md @@ -850,6 +850,28 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +## Axiom of Choice + +### Relation Form + +For any relation $R$ there exists a function $F \subseteq R$ with $\mathop{\text{dom}}F = \mathop{\text{dom}}R$. + +%%ANKI +Basic +What is AoC an acronym for? +Back: The **A**xiom **o**f **C**hoice. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What does the Axiom of Choice (relation form) state? +Back: For any relation $R$ there exists a function $F \subseteq R$ with $\mathop{\text{dom}}F = \mathop{\text{dom}}R$. +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). diff --git a/notes/set/relations.md b/notes/set/relations.md index aa67912..88c250c 100644 --- a/notes/set/relations.md +++ b/notes/set/relations.md @@ -95,6 +95,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +Is $\varnothing$ a relation? +Back: Yes. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + A **relation** $R$ is a set of ordered pairs. The **domain** of $R$ ($\mathop{\text{dom}}{R}$), the **range** of $R$ ($\mathop{\text{ran}}{R}$), and the **field** of $R$ ($\mathop{\text{fld}}{R}$) is defined as: * $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$