From 65517ee3f80243b13faef87afd4f3b3f1a78eb00 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Wed, 10 Jul 2024 20:47:37 -0600 Subject: [PATCH] Image operations and b-nf class characterization. --- .../plugins/obsidian-to-anki-plugin/data.json | 14 +- notes/_journal/2024-07-10.md | 12 ++ notes/_journal/{ => 2024-07}/2024-07-08.md | 0 notes/_journal/2024-07/2024-07-09.md | 9 + notes/lambda-calculus/beta-reduction.md | 160 ++++++++++++++++-- notes/logic/quantification.md | 16 +- notes/set/functions.md | 65 ++++++- notes/set/index.md | 8 +- 8 files changed, 250 insertions(+), 34 deletions(-) create mode 100644 notes/_journal/2024-07-10.md rename notes/_journal/{ => 2024-07}/2024-07-08.md (100%) create mode 100644 notes/_journal/2024-07/2024-07-09.md diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 865760a..88da64d 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -194,7 +194,7 @@ "binary/index.md": "9089c6f0e86a0727cd03984f51350de0", "_journal/2024-02-09.md": "a798d35f0b2bd1da130f7ac766166109", "c/types.md": "cf3e66e5aee58a94db3fdf0783908555", - "logic/quantification.md": "313cb37b33dfe7604ba4b8d1db4fb90f", + "logic/quantification.md": "a83c424bf9f0bc675e9bd09f8118317b", "c/declarations.md": "2de27f565d1020819008ae80593af435", "algorithms/sorting/bubble-sort.md": "872fb23e41fb3ac36e8c46240e9a027f", "_journal/2024-02-10.md": "562b01f60ea36a3c78181e39b1c02b9f", @@ -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": "43b219df1822f002fdac63aa6d1c8f9a", + "set/index.md": "9444d7f3660f1b308d268d5833997737", "set/graphs.md": "55298be7241906cb6b61673cf0a2e709", "_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb", "_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391", @@ -534,10 +534,10 @@ "_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": "1a09c0a0f505c5f551a04f0595971d56", + "set/functions.md": "4fd3388fb21c77e96c6cfb703f3ed153", "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4", "_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307", - "lambda-calculus/beta-reduction.md": "5532f9beec9d265724a8d205326bcf67", + "lambda-calculus/beta-reduction.md": "5386403713b42ee1831d15c84de133ba", "_journal/2024-06-16.md": "ded6ab660ecc7c3dce3afd2e88e5a725", "_journal/2024-06/2024-06-15.md": "c3a55549da9dfc2770bfcf403bf5b30b", "_journal/2024-06-17.md": "63df6757bb3384e45093bf2b9456ffac", @@ -585,7 +585,11 @@ "_journal/2024-07-07.md": "9ee2d5007c34cc7ff681f3d9e998eca4", "_journal/2024-07/2024-07-06.md": "2b794e424985f0e7d4d899163ce5733c", "_journal/2024-07-08.md": "03ed5604e680ac9742ee99ae4b1eee8b", - "_journal/2024-07/2024-07-07.md": "9ee2d5007c34cc7ff681f3d9e998eca4" + "_journal/2024-07/2024-07-07.md": "9ee2d5007c34cc7ff681f3d9e998eca4", + "_journal/2024-07-09.md": "00c357e9cfac6de17825b02fdbd00c80", + "_journal/2024-07/2024-07-08.md": "03ed5604e680ac9742ee99ae4b1eee8b", + "_journal/2024-07-10.md": "2bb3db1f506f4ec7726cb5f2ed2daf24", + "_journal/2024-07/2024-07-09.md": "00c357e9cfac6de17825b02fdbd00c80" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-07-10.md b/notes/_journal/2024-07-10.md new file mode 100644 index 0000000..d7f7cf6 --- /dev/null +++ b/notes/_journal/2024-07-10.md @@ -0,0 +1,12 @@ +--- +title: "2024-07-10" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Notes on an alternative classification for $\beta\text{-nf}$. +* Small notes on the difference of images. \ No newline at end of file diff --git a/notes/_journal/2024-07-08.md b/notes/_journal/2024-07/2024-07-08.md similarity index 100% rename from notes/_journal/2024-07-08.md rename to notes/_journal/2024-07/2024-07-08.md diff --git a/notes/_journal/2024-07/2024-07-09.md b/notes/_journal/2024-07/2024-07-09.md new file mode 100644 index 0000000..f8050e7 --- /dev/null +++ b/notes/_journal/2024-07/2024-07-09.md @@ -0,0 +1,9 @@ +--- +title: "2024-07-09" +--- + +- [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/lambda-calculus/beta-reduction.md b/notes/lambda-calculus/beta-reduction.md index d7c7ce4..c49f773 100644 --- a/notes/lambda-calculus/beta-reduction.md +++ b/notes/lambda-calculus/beta-reduction.md @@ -210,7 +210,7 @@ END%% ## Normal Form -A term $Q$ which contains no $\beta$-redexes is called a **$\beta$-normal form** (or a **term in $\beta$-normal form** or just a **$\beta$-nf**). The class of all $\beta$-normal forms is called $\beta$-nf or $\lambda\beta$-nf. If a term $P$ $\beta$-reduces to a term $Q$ in $\beta$-nf, then $Q$ is called a **$\beta$-normal form of $P$**. +A term $Q$ which contains no $\beta$-redexes is called a **$\beta$-normal form** (or a **term in $\beta$-normal form** or just a **$\beta\text{-nf}$**). The class of all $\beta$-normal forms is called $\beta\text{-nf}$ or $\lambda\beta\text{-nf}$. If a term $P$ $\beta$-reduces to a term $Q$ in $\beta\text{-nf}$, then $Q$ is called a **$\beta$-normal form of $P$**. %%ANKI Basic @@ -246,14 +246,14 @@ END%% %%ANKI Cloze -The class of {all $\beta$-normal forms} is called {$\beta$-nf/$\lambda\beta$-nf}. +The class of {all $\beta$-normal forms} is called {$\beta\text{-nf}$/$\lambda\beta\text{-nf}$}. 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 -What ambiguity does term "$\beta$-nf" introduce? +What ambiguity does term "$\beta\text{-nf}$" introduce? Back: It refers to a specific $\beta$-normal form or the class of $\beta$-normal forms. 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). @@ -262,23 +262,23 @@ END%% %%ANKI Basic What does it mean for term $Q$ to be a $\beta$-normal form of term $P$? -Back: $P$ $\beta$-reduces to a term $Q$ in $\beta$-nf. +Back: $P$ $\beta$-reduces to a term $Q$ in $\beta\text{-nf}$. 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 -How is the class $\beta$-nf alternatively denoted? -Back: As $\lambda\beta$-nf. +How is the class $\beta\text{-nf}$ alternatively denoted? +Back: As $\lambda\beta\text{-nf}$. 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 -How is the class $\lambda\beta$-nf alternatively denoted? -Back: As $\beta$-nf. +How is the class $\lambda\beta\text{-nf}$ alternatively denoted? +Back: As $\beta\text{-nf}$. 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%% @@ -325,20 +325,36 @@ END%% %%ANKI Basic -Why isn't $x(\lambda u. uv)$ in $\beta$-normal form? -Back: N/A. It is. +Is $a(\lambda u. uv)x$ 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 +*Why* is $x(\lambda u. uv)$ in $\beta$-normal form? +Back: It contains no $\beta$-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%% %%ANKI Basic -Why isn't $(\lambda u. uv)x$ in $\beta$-normal form? +*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%% +%%ANKI +Basic +*Why* is $a(\lambda u. uv)x$ in $\beta$-normal form? +Back: With parentheses, $(a(\lambda u. uv))x$ clearly contains no $\beta$-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%% + %%ANKI Basic Let $P \,\triangleright_\beta\, Q$. How do $FV(P)$ and $FV(Q)$ relate to one another? @@ -363,11 +379,127 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi END%% +As an alternative characterization, the class $\beta\text{-nf}$ is the smallest class such that + +* all atoms are in $\beta\text{-nf}$; +* $M, N \in \beta\text{-nf} \Rightarrow aMN \in \beta\text{-nf}$ for all atoms $a$; +* $M \in \beta\text{-nf} \Rightarrow \lambda x. M \in \beta\text{-nf}$ + +%%ANKI +Basic +What proposition explains how atoms relate to the definition of $\beta\text{-nf}$? +Back: All atoms are in $\beta\text{-nf}$. +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 +What proposition explains how applications relate to the definition of $\beta\text{-nf}$? +Back: For all atoms $a$, if $M, N \in \beta\text{-nf}$, then $aMN \in \beta\text{-nf}$. +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 +Given atom $a$ and $M \in \beta\text{-nf}$, what application is in $\beta\text{-nf}$? +Back: $aM$ +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 +$M, N \in \beta\text{-nf}$ implies what application is in $\beta\text{-nf}$? +Back: $aMN$ for any atom $a$. +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 +Given $M, N \in \beta\text{-nf}$, when is $MN \in \beta\text{-nf}$? +Back: When $M$ is not an abstraction. +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 +Given $M, N \in \beta\text{-nf}$, when is $MN \not\in \beta\text{-nf}$? +Back: When $M$ is an abstraction. +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 +What proposition explains how abstractions relate to the definition of $\beta\text{-nf}$? +Back: If $M \in \beta\text{-nf}$, then $\lambda x. M \in \beta\text{-nf}$. +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 +$M \in \beta\text{-nf}$ implies what abstraction is in $\beta\text{-nf}$? +Back: $\lambda x. M$ +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 +Given atom $a$, if $M \equiv aM_1\ldots M_n$ and $M \,\triangleright_\beta\, N$, what form does $N$ have? +Back: $aN_1\ldots N_n$ where $M_i \,\triangleright_\beta\, N_i$ for $i = 1, \ldots, n$. +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 +Given atom $a$, if $M \equiv aM_1\ldots M_n$ and $M \,\triangleright_\beta\, N$, *why* does $N$ have form $aN_1\ldots N_n$? +Back: Since $M \equiv ((\cdots((aM_1)M_2)\cdots)M_n)$, every $\beta$-redex must be in an $M_i$. +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 +What does it mean for a $\lambda$-term to *be* a $\beta\text{-nf}$? +Back: The $\lambda$-term contains no $\beta$-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%% + +%%ANKI +Basic +What does it mean for a $\lambda$-term to *have* a $\beta\text{-nf}$? +Back: The $\lambda$-term can be $\beta$-reduced into a term in $\beta\text{-nf}$. +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 +Suppose $[N/x]M$ is a $\beta\text{-nf}$. Is $M$ a $\beta\text{-nf}$? +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 +Suppose $[N/x]M$ has a $\beta\text{-nf}$. Does $M$ have a $\beta\text{-nf}$? +Back: Not necessarily. +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%% + ## Church-Rosser Theorem -If $P \,\triangleright_\beta\, M$ and $P \,\triangleright_\beta\, N$, then there exists a term $T$ such that $M \,\triangleright_\beta\, T$ and $N \,\triangleright_\beta\, T$. - -As an immediate corollary, if $P$ has a $\beta$-normal form then it it is unique modulo $\equiv_\alpha$. +If $P \,\triangleright_\beta\, M$ and $P \,\triangleright_\beta\, N$, then there exists a term $T$ such that $M \,\triangleright_\beta\, T$ and $N \,\triangleright_\beta\, T$. As an immediate corollary, if $P$ has a $\beta$-normal form then it it is unique modulo $\equiv_\alpha$. %%ANKI Basic diff --git a/notes/logic/quantification.md b/notes/logic/quantification.md index d045bf6..4818434 100644 --- a/notes/logic/quantification.md +++ b/notes/logic/quantification.md @@ -232,7 +232,7 @@ END%% %%ANKI Basic -Which identifiers in the following are bound? $$\exists x, P(x) \land P(y)$$ +Which identifiers in the following are bound? $\exists x, P(x) \land P(y)$ Back: Just $x$. Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. @@ -240,7 +240,7 @@ END%% %%ANKI Basic -Which identifiers in the following are free? $$\exists x, P(x) \land P(y)$$ +Which identifiers in the following are free? $\exists x, P(x) \land P(y)$ Back: Just $y$. Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. @@ -248,12 +248,20 @@ END%% %%ANKI Basic -How is the following rewritten in PNF? $$(\exists x, P(x)) \land (\exists y, P(y))$$ -Back: $\exists x \;y, P(x) \land P(y)$ +How is the following rewritten in PNF? $(\exists x, P(x)) \land (\exists y, Q(y))$ +Back: $\exists x \;y, P(x) \land Q(y)$ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. END%% +%%ANKI +Basic +How is the following rewritten in PNF? $(\exists x, P(x)) \land (\forall y, Q(y))$ +Back: N/A. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + ## Bibliography * Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. diff --git a/notes/set/functions.md b/notes/set/functions.md index dbb66b6..0458260 100644 --- a/notes/set/functions.md +++ b/notes/set/functions.md @@ -1271,12 +1271,15 @@ The following holds for any sets $F$, $A$, $B$, and $\mathscr{A}$: * The image of unions is the union of the images: * $F[\![\bigcup\mathscr{A}]\!] = \bigcup\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$ * The image of intersections is a subset of the intersection of images: - * $F[\![\bigcap \mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$ + * $F[\![\bigcap \mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$ for $\mathscr{A} \neq \varnothing$ + * Equality holds if $F$ is single-rooted. +* The image of a difference includes the difference of the images: + * $F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]$ * Equality holds if $F$ is single-rooted. %%ANKI Basic -How does the image of unions relate to the union of images? +How does the image of unions relate to the union of the images? Back: They are equal. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). @@ -1284,7 +1287,7 @@ END%% %%ANKI Basic -How does the union of images relate to the images of unions? +How does the union of images relate to the images of the unions? Back: They are equal. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). @@ -1324,7 +1327,7 @@ END%% %%ANKI Basic -How does the image of intersections relate to the intersection of images? +How does the image of intersections relate to the intersection of the images? Back: The former is a subset of the latter. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). @@ -1332,7 +1335,7 @@ END%% %%ANKI Basic -How does the intersection of images relate to the image of intersections? +How does the intersection of images relate to the image of the intersections? Back: The latter is a subset of the former. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). @@ -1380,8 +1383,8 @@ END%% %%ANKI Basic -*Why* is the following identity intuitively true? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$ -Back: $A \cap B$ could be empty but $F[\![A]\!] \cap F[\![B]\!]$ could be nonempty. +What $\varnothing$-based example is used to show the following is intuitively true? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$ +Back: $A$ and $B$ might be disjoint even if $F[\![A]\!]$ and $F[\![B]\!]$ are not. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% @@ -1402,6 +1405,54 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +How does the image of differences relate to the difference of the images? +Back: The latter is a subset of the former. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How does the difference of images relate to the image of the differences? +Back: The former is a subset of the latter. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What $\varnothing$-based example is used to show the following is intuitively true? $$F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]$$ +Back: $F[\![A]\!]$ and $F[\![B]\!]$ might be the same sets even if $A \neq B$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What condition on set $F$ makes the following true? $F[\![A - B]\!] \subseteq F[\![A]\!] - F[\![B]\!]$ +Back: $F$ is single-rooted. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What condition on set $F$ makes the following true? $F[\![A - B]\!] = F[\![A]\!] - F[\![B]\!]$ +Back: $F$ is single-rooted. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What condition on set $F$ makes the following true? $F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]$ +Back: N/A. This is always true. +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). diff --git a/notes/set/index.md b/notes/set/index.md index 2dbec4f..50058a3 100644 --- a/notes/set/index.md +++ b/notes/set/index.md @@ -251,8 +251,8 @@ END%% %%ANKI Basic -How many members are in set $\{P(y) \mid y \in B\}$? -Back: As many as the number of unique $P(y)$ for each $y \in B$. +Given function $P$, how is set $\{P(y) \mid y \in B\}$ more compactly denoted? +Back: $P[\![B]\!]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% @@ -307,8 +307,8 @@ END%% %%ANKI Basic -How do you rewrite $\{A \mid A \in B\}$ with an existential in the entrance requirement? -Back: $\{v \mid A \in B \land v = A\}$ +How is $\{A \mid A \in B\}$ rewritten with an existential in the entrance requirement? +Back: $\{v \mid \exists A \in B \land v = A\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%%