Fixup flashcards. Cartesian products.
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"combinatorics/inclusion-exclusion.md": "c27b49ee03cc5ee854d0e8bd12a1d505",
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@ -249,7 +249,7 @@
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"algorithms/sorting/merge-sort.md": "6506483f7df6507cee0407bd205dbedd",
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"_journal/2024-02/2024-02-23.md": "0aad297148e8cc4058b48b7e45787ca7",
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"set/axioms.md": "063955bf19c703e9ad23be2aee4f1ab7",
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"_journal/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5",
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"_journal/2024-05-16.md": "580c7ec61ec56be92fa8d6affcf0a5f6",
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@ -475,9 +475,9 @@
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"_journal/2024-05/2024-05-26.md": "abe84b5beae74baa25501c818e64fc95",
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"algebra/boolean.md": "ee41e624f4d3d3aca00020d9a9ae42c8",
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"git/merge-conflicts.md": "af3603c7a8fde60a1982c0950d209b77",
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"git/merge-conflicts.md": "761ad6137ec51d3877f7d5b3615ca5cb",
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"_journal/2024-05-28.md": "0f6aeb5ec126560acdc2d8c5c6570337",
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"_journal/2024-05-29.md": "aee3f3766659789d7dfb63dd247844cc",
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"_journal/2024-06/2024-06-07.md": "c6bfc4c1e5913d23ea7828a23340e7d3",
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"lambda-calculus/alpha-conversion.md": "c0d40271a14b1f44b937de7791ca089b",
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"x86-64/instructions/arithmetic.md": "271218d855e7291f119f96e91f582738",
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"x86-64/instructions/index.md": "72c19067e938ab39ea51d25d6ac2bad9",
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"_journal/2024-06-09.md": "935b3ddf65c51e680ac5c000c7e380af",
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"_journal/2024-06/2024-06-08.md": "9e1ebc8882a395b96ca765ad5c982d68"
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"_journal/2024-06/2024-06-08.md": "9e1ebc8882a395b96ca765ad5c982d68",
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"_journal/2024-06-10.md": "84d27300b97c8544ab4ec68b06edd824",
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"_journal/2024-06/2024-06-09.md": "4c336a39775846b416aa73278435065f",
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"_journal/2024-06-11.md": "4988503ee98aa4fe32d5eeaea2cc65a5",
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"_journal/2024-06/2024-06-10.md": "1fe3a8beb03b1cc9af188b85933339e4"
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},
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"fields_dict": {
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"Basic": [
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---
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title: "2024-06-11"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [x] OGS
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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---
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title: "2024-06-10"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [x] OGS
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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* Various notes on the Cartesian product.
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@ -87,6 +87,46 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1716396060620-->
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END%%
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%%ANKI
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Basic
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Is the Cartesian product commutative?
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Back: No.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881694-->
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END%%
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%%ANKI
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Basic
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*Why* isn't the Cartesian product commutative?
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Back: Because the Cartesian product comprises of *ordered* pairs.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881698-->
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END%%
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%%ANKI
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Basic
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Suppose $A \neq \varnothing$ and $B \neq \varnothing$. When does $A \times B = B \times A$?
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Back: When $A = B$.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881702-->
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END%%
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%%ANKI
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Basic
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Suppose $A \neq \varnothing$ and $A \neq B$. When does $A \times B = B \times A$?
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Back: When $B = \varnothing$.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881705-->
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END%%
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%%ANKI
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Basic
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Under what two conditions is $A \times B = B \times A$?
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Back: $A = B$ or either set is the empty set.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881709-->
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END%%
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### Associative Laws
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For any sets $A$ and $B$, $$\begin{align*} A \cup (B \cup C) & = (A \cup B) \cup C \\ A \cap (B \cap C) & = (A \cap B) \cap C \end{align*}$$
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@ -115,6 +155,22 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1716396060625-->
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END%%
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%%ANKI
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Basic
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Is the Cartesian product associative?
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Back: No.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881712-->
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END%%
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%%ANKI
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Basic
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*Why* isn't the Cartesian product associative?
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Back: The association of parentheses defines the nesting of the ordered pairs.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881715-->
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END%%
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### Distributive Laws
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For any sets $A$, $B$, and $C$, $$\begin{align*} A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) & = (A \cup B) \cap (A \cup C) \end{align*}$$
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@ -253,6 +309,94 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1717372322287-->
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END%%
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For any sets $A$, $B$, and $C$, $$\begin{align*} A \times (B \cap C) & = (A \times B) \cap (A \times C) \\ A \times (B \cup C) & = (A \times B) \cup (A \times C) \\ A \times (B - C) & = (A \times B) - (A \times C) \end{align*}$$
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%%ANKI
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Basic
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Which algebra of sets operators is the Cartesian product distributive over?
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Back: $\cap$, $\cup$, and $-$
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881718-->
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END%%
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%%ANKI
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Basic
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What distributivity rule is satisfied by $\cap$ and $\times$?
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Back: $A \times (B \cap C) = (A \times B) \cap (A \times C)$
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881723-->
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END%%
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%%ANKI
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Cloze
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The Cartesian product satisfies distributivity: {$A \times (B \cap C)$} $=$ {$(A \times B) \cap (A \times C)$}.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881726-->
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END%%
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%%ANKI
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Basic
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What distributivity rule is satisfied by $\cup$ and $\times$?
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Back: $A \times (B \cup C) = (A \times B) \cup (A \times C)$
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881731-->
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END%%
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%%ANKI
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Cloze
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The Cartesian product satisfies distributivity: {$A \times (B \cup C)$} $=$ {$(A \times B) \cup (A \times C)$}.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881735-->
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END%%
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%%ANKI
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Basic
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What distributivity rule is satisfied by $-$ and $\times$?
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Back: $A \times (B - C) = (A \times B) - (A \times C)$
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881742-->
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END%%
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%%ANKI
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Cloze
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The Cartesian product satisfies distributivity: {$A \times (B - C)$} $=$ {$(A \times B) - (A \times C)$}.
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881752-->
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END%%
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In addition, $$\begin{align*} A \times \bigcup \mathscr{B} & = \bigcup\, \{A \times X \mid X \in \mathscr{B}\} \\ A \times \bigcap \mathscr{B} & = \bigcap\, \{A \times X \mid X \in \mathscr{B}\} \end{align*}$$
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%%ANKI
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Basic
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What is the generalization of identity $A \times (B \cup C) = (A \times B) \cup (A \times C)$?
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Back: $A \times \bigcup \mathscr{B} = \bigcup\, \{A \times X \mid X \in \mathscr{B}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718069881759-->
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END%%
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%%ANKI
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Basic
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What is the specialization of identity $A \times \bigcap \mathscr{B} = \bigcap\, \{A \times X \mid X \in \mathscr{B}\}$?
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Back: $A \times (B \cap C) = (A \times B) \cap (A \times C)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718069881766-->
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END%%
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%%ANKI
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Basic
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What is the generalization of identity $A \times (B \cap C) = (A \times B) \cap (A \times C)$?
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Back: $A \times \bigcap \mathscr{B} = \bigcap\, \{A \times X \mid X \in \mathscr{B}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718069881773-->
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END%%
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%%ANKI
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Basic
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What is the specialization of identity $A \times \bigcup \mathscr{B} = \bigcup\, \{A \times X \mid X \in \mathscr{B}\}$?
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Back: $A \times (B \cup C) = (A \times B) \cup (A \times C)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718069881779-->
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END%%
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### De Morgan's Laws
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For any sets $A$, $B$, and $C$, $$\begin{align*} C - (A \cup B) & = (C - A) \cap (C - B) \\ C - (A \cap B) & = (C - A) \cup (C - B) \end{align*}$$
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@ -458,6 +602,18 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1717073536985-->
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END%%
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In addition,
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* $A \subseteq B \Rightarrow A \times C \subseteq B \times C$
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%%ANKI
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Basic
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What monotonicity property does the Cartesian product satisfy?
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Back: $A \subseteq B \Rightarrow A \times C \subseteq B \times C$
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Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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<!--ID: 1718069881786-->
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END%%
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### Antimonotonicity
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Let $A$, $B$, and $C$ be arbitrary sets. Then
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%%ANKI
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Cloze
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The C standard sets {1:lower bounds} on data type ranges, but does not set {1:upper bounds} (except with fixed-size types).
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{1:Lower} bounds are defined on data type's numeric ranges, but not {1:upper} (except for {2:fixed-size} types).
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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<!--ID: 1707493017244-->
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END%%
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@ -327,14 +327,6 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
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<!--ID: 1708434662157-->
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END%%
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%%ANKI
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Basic
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What isomorphism shows finite sets have an equal number of even-sized subsets as odd-sized subsets?
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Back: The one-to-one relationship between subsets and bit strings.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708434662160-->
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END%%
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%%ANKI
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Basic
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Given finite set $A$, how many subsets of even cardinality does $A$ have?
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@ -44,7 +44,7 @@ END%%
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|||
|
||||
%%ANKI
|
||||
Basic
|
||||
In a `git merge`, what changes are between `<<<<<<<` and `=======`?
|
||||
In a `git merge`, what changes are after `<<<<<<<` and before `=======`?
|
||||
Back: The changes present on the current branch.
|
||||
Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014).
|
||||
<!--ID: 1716804846990-->
|
||||
|
|
@ -52,7 +52,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
In a `git merge`, what changes are between `>>>>>>>` and `=======`?
|
||||
In a `git merge`, what changes are after `>>>>>>>` and before `=======`?
|
||||
Back: N/A.
|
||||
Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014).
|
||||
<!--ID: 1716804846993-->
|
||||
|
|
@ -60,7 +60,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
In a `git merge`, what changes are between `=======` and `>>>>>>>`?
|
||||
In a `git merge`, what changes are after `=======` and before `>>>>>>>`?
|
||||
Back: The changes present on the branch being merged into `HEAD`.
|
||||
Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014).
|
||||
<!--ID: 1716804846999-->
|
||||
|
|
@ -68,7 +68,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
In a `git merge`, what changes are between `=======` and `<<<<<<<`?
|
||||
In a `git merge`, what changes are after `=======` and before `<<<<<<<`?
|
||||
Back: N/A.
|
||||
Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014).
|
||||
<!--ID: 1716804846996-->
|
||||
|
|
|
|||
|
|
@ -416,7 +416,7 @@ END%%
|
|||
%%ANKI
|
||||
Basic
|
||||
How is "occurs in" recursively defined for application?
|
||||
Back: If $P$ occurs in $M$ or $N$, then $P$ occurs in $(MN)$.
|
||||
Back: $P$ occurs in $(MN)$ if $P$ occurs in $M$ or $P$ occurs in $N$.
|
||||
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).
|
||||
<!--ID: 1716743248122-->
|
||||
END%%
|
||||
|
|
@ -424,7 +424,7 @@ END%%
|
|||
%%ANKI
|
||||
Basic
|
||||
How is "occurs in" recursively defined for abstraction?
|
||||
Back: If $P$ occurs in $M$ or $P$ is $x$, then $P$ occurs in $(\lambda x. M)$.
|
||||
Back: $P$ occurs in $(\lambda x. M)$ if $P$ occurs in $M$ or $P$ is $x$.
|
||||
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).
|
||||
<!--ID: 1716743248124-->
|
||||
END%%
|
||||
|
|
@ -618,7 +618,7 @@ END%%
|
|||
%%ANKI
|
||||
Basic
|
||||
Which specific occurrences are bound in $\lambda x. x(\lambda y. yz)$?
|
||||
Back: Each $x$ and each $y$.
|
||||
Back: Both $x$s and both $y$s.
|
||||
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).
|
||||
<!--ID: 1716745016028-->
|
||||
END%%
|
||||
|
|
|
|||
|
|
@ -51,9 +51,9 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is $\langle x, y \rangle$ usually defined?
|
||||
How is $\langle x, y \rangle$ most commonly defined?
|
||||
Back: As $\{\{x\}, \{x, y\}\}$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
||||
<!--ID: 1717678753120-->
|
||||
END%%
|
||||
|
||||
|
|
@ -125,7 +125,7 @@ END%%
|
|||
Basic
|
||||
Why is the Cartesian product named the way it is?
|
||||
Back: It is named after René Descartes.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
||||
<!--ID: 1717679397836-->
|
||||
END%%
|
||||
|
||||
|
|
@ -146,4 +146,5 @@ END%%
|
|||
|
||||
## Bibliography
|
||||
|
||||
* “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
|
||||
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
|
|
@ -215,8 +215,8 @@ END%%
|
|||
Basic
|
||||
Evaluate `int t = a + b` for `int` `a` and `b`. When is `CF` set?
|
||||
Back: When `(unsigned) t < (unsigned) a`.
|
||||
Tags: c17
|
||||
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
||||
Tags: c17
|
||||
<!--ID: 1717941416597-->
|
||||
END%%
|
||||
|
||||
|
|
@ -224,8 +224,8 @@ END%%
|
|||
Basic
|
||||
Evaluate `int t = a + b` for `int` `a` and `b`. When is `ZF` set?
|
||||
Back: When `t == 0`.
|
||||
Tags: c17
|
||||
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
||||
Tags: c17
|
||||
<!--ID: 1717941416601-->
|
||||
END%%
|
||||
|
||||
|
|
@ -233,8 +233,8 @@ END%%
|
|||
Basic
|
||||
Evaluate `int t = a + b` for `int` `a` and `b`. When is `SF` set?
|
||||
Back: When `t < 0`.
|
||||
Tags: c17
|
||||
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
||||
Tags: c17
|
||||
<!--ID: 1717941416605-->
|
||||
END%%
|
||||
|
||||
|
|
@ -242,8 +242,8 @@ END%%
|
|||
Basic
|
||||
Evaluate `int t = a + b` for `int` `a` and `b`. When is `OF` set?
|
||||
Back: When `(t <= 0 && a > 0 && b > 0) || (t >= 0 && a < 0 && b < 0)`.
|
||||
Tags: c17
|
||||
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
||||
Tags: c17
|
||||
<!--ID: 1717941416609-->
|
||||
END%%
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue