From 87ca0829b2802b72abb8b43b13d7a056a2aa7a69 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Sat, 22 Jun 2024 18:48:45 -0600 Subject: [PATCH] More beta-reduction notes and set theory definitions. --- .../plugins/obsidian-to-anki-plugin/data.json | 4 +- notes/_journal/2024-06-22.md | 3 +- notes/lambda-calculus/beta-reduction.md | 34 ++- notes/set/functions.md | 255 +++++++++++++++++- 4 files changed, 291 insertions(+), 5 deletions(-) diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 6de17e4..d2b265d 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -533,10 +533,10 @@ "_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": "9fc813971de5fdda7aaac0cf91a721ad", + "set/functions.md": "29fbc5fe127487649cd260aa9e290a3a", "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4", "_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307", - "lambda-calculus/beta-reduction.md": "5ea9b74d57074c699086db060fb86d50", + "lambda-calculus/beta-reduction.md": "e8b7fbca6c0395314ecbaf93a24e653b", "_journal/2024-06-16.md": "ded6ab660ecc7c3dce3afd2e88e5a725", "_journal/2024-06/2024-06-15.md": "c3a55549da9dfc2770bfcf403bf5b30b", "_journal/2024-06-17.md": "63df6757bb3384e45093bf2b9456ffac", diff --git a/notes/_journal/2024-06-22.md b/notes/_journal/2024-06-22.md index 92f26f5..92b0a03 100644 --- a/notes/_journal/2024-06-22.md +++ b/notes/_journal/2024-06-22.md @@ -9,4 +9,5 @@ title: "2024-06-22" - [ ] Korean (Read 1 Story) * Notes on [[beta-reduction#Normal Form|β-normal forms]]. -* Very basic notes on [[ars|abstract rewriting systems]]. \ No newline at end of file +* Very basic notes on [[ars|abstract rewriting systems]]. +* Additional set theory definitions ([[functions#Restrictions|restrictions]] and [[functions#Images|images]]). \ No newline at end of file diff --git a/notes/lambda-calculus/beta-reduction.md b/notes/lambda-calculus/beta-reduction.md index 9a39991..838e2fd 100644 --- a/notes/lambda-calculus/beta-reduction.md +++ b/notes/lambda-calculus/beta-reduction.md @@ -260,12 +260,44 @@ END%% %%ANKI Basic -How many contractions can make up a $\beta$-reduction? +How many contractions make up a $\beta$-reduction? Back: Zero or more. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). END%% +%%ANKI +Basic +Is $x(\lambda u. uv)$ in $\beta$-normal form? +Back: Yes. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Is $(\lambda u. uv)x$ in $\beta$-normal form? +Back: No. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Why isn't $x(\lambda u. uv)$ in $\beta$-normal form? +Back: N/A. It is. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Why isn't $(\lambda u. uv)x$ in $\beta$-normal form? +Back: Because $(\lambda u. uv)x$ is a $\beta$-redex. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + ## Bibliography * Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). \ No newline at end of file diff --git a/notes/set/functions.md b/notes/set/functions.md index f35b535..aee0cda 100644 --- a/notes/set/functions.md +++ b/notes/set/functions.md @@ -440,7 +440,7 @@ END%% ## Inverses -Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}.$$ +Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}$$ %%ANKI Basic What kind of mathematical object does the inverse operation apply to? @@ -617,6 +617,22 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +How is set $\{\langle u, v \rangle \mid vAu\}$ more simply denoted? +Back: $A^{-1}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What does $\varnothing^{-1}$ evalute to? +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 $$F \circ G = \{\langle u, v \rangle \mid \exists t, uGt \land tFv \}$$ @@ -676,6 +692,243 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +How is set $\{\langle u, v \rangle \mid \exists t, uBt \land tAv \}$ more simply denoted? +Back: $A \circ B$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $F$ be an arbitrary set. What is $F \circ \varnothing$? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $F$ be an arbitrary set. What is $\varnothing \circ F$? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +## Restrictions + +Let $F$ and $A$ be arbitrary sets. The **restriction of $F$ to $A$** is the set $$F \restriction A = \{\langle u, v \rangle \mid uFv \land u \in A\}$$ + +%%ANKI +Basic +What kind of mathematical object does the restriction operation apply to? +Back: Sets. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +$F \restriction A$ is the restriction of $F$ {to} $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What kind of mathematical object does the restriction operation emit? +Back: Relations. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the "arity" of the restriction operation in set theory? +Back: $2$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is the restriction of $F$ to $A$ denoted? +Back: $F \restriction A$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is the restriction of $F$ to $A$ defined? +Back: $F \restriction A = \{\langle u, v \rangle \mid uFv \land u \in A\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider function $F \colon A \rightarrow B$. How does $\mathop{\text{dom}}F$ relate to $\mathop{\text{dom}}(F \restriction A)$? +Back: They are equal. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider function $F \colon A \rightarrow B$. How does $\mathop{\text{ran}}F$ relate to $\mathop{\text{ran}}(F \restriction A)$? +Back: They are equal. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider function $F \colon A \rightarrow B$ and set $C \subseteq A$. How does $\mathop{\text{dom}}F$ relate to $\mathop{\text{dom}}(F \restriction C)$? +Back: $\mathop{\text{dom}}(F \restriction C) \subseteq \mathop{\text{dom}}F$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is $F \restriction A$ pronounced? +Back: The restriction of $F$ to $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Consider function $F \colon A \rightarrow B$ and set $C \subseteq A$. How does $\mathop{\text{ran}}F$ relate to $\mathop{\text{ran}}(F \restriction C)$? +Back: $\mathop{\text{ran}}(F \restriction C) \subseteq \mathop{\text{ran}}F$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is set $\{\langle u, v \rangle \mid uAv \land u \in B\}$ more simply denoted? +Back: $A \restriction B$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $F$ be an arbitrary set. What is $F \restriction \varnothing$? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +## Images + +Let $F$ and $A$ be sets. Then the **image of $F$ under $A$** is $$F[\![A]\!] = \{v \mid \exists u \in A, uFv\}$$ + +%%ANKI +Basic +What kind of mathematical object does the image operation apply to? +Back: Sets. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What kind of mathematical object does the image operation emit? +Back: Sets. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +$F[\![A]\!]$ is the image of $F$ {under} $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the "arity" of the image operation in set theory? +Back: $2$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is the image of $F$ under $A$ denoted? +Back: $F[\![A]\!]$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is the image of $F$ under $A$ defined? +Back: $F[\![A]\!] = \{v \mid \exists u \in A, uFv\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is the image of $F$ under $A$ defined in terms of restrictions? +Back: $F[\![A]\!] = \mathop{\text{ran}}(F \restriction A)$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is set $\{v \mid \exists u \in B, uAv\}$ more simply denoted? +Back: $A[\![B]\!]$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Enderton says "multiple-valued functions" are actually what? +Back: Relations. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Enderton says "multiple-valued functions" are actually what? +Back: Relations. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). +END%% + +%%ANKI +Basic +Enderton says "$F^{-1}(9) = \pm 3$" is preferably written in what way? +Back: $F^{-1}[\![\{9\}]\!] = \{-3, 3\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +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 $F[\![\varnothing]\!]$? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ## Bibliography * “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).