diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index a64a3c1..9da414e 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -350,7 +350,7 @@ "_journal/2024-02-23.md": "219ce9ad15a8733edd476c97628b71fd", "_journal/2024-02/2024-02-22.md": "312e55d57868026f6e80f7989a889c2b", "c17/strings.md": "b021b6f18f865aa89b3088907cdaae94", - "c17/index.md": "4b5a563b8079ea6afb9c57648a2b0124", + "c17/index.md": "78e0ebc5a91d58b297b25346a654d398", "c17/escape-sequences.md": "a8b99070336878b4e8c11e9e4525a500", "c17/declarations.md": "9dbaea14e28d00704cfd0b027efe8c97", "algorithms/sorting/merge-sort.md": "6506483f7df6507cee0407bd205dbedd", @@ -413,7 +413,7 @@ "_journal/2024-03/2024-03-15.md": "e54b2513beac5f46313b4c37622adf39", "_journal/2024-03-17.md": "72e99c7630085aee2c7f340a06b5ada7", "_journal/2024-03/2024-03-16.md": "ab7629c24ebe70838072cf6acec47cb0", - "encoding/floating-point.md": "23219ba579de6bb7ccf0cccc2329e183", + "encoding/floating-point.md": "d19e3f992bf2e073d2049fb0973c89bd", "_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f", "_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b", "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53", @@ -878,7 +878,7 @@ "_journal/2024-09/2024-09-26.md": "2d3e8325e7ab63168c460f18e7aa1afc", "_journal/2024-09-28.md": "7726baed125a2561def07dcaf48bf5a0", "_journal/2024-09/2024-09-27.md": "d788fa04c029009f42387317c549d93e", - "encoding/binary.md": "0b9beb6913906aa2523d8ab193c67f67", + "encoding/binary.md": "a7cf4467f1edc6a27869b958cf9f1dba", "_journal/2024-09-29.md": "232733c9ad7ebd89e8834cd61e1536d7", "_journal/2024-09/2024-09-28.md": "1b47792313acf09b1ae768d5918df703", "_journal/2024-10-01.md": "2547ea281e11e2bc728d2b21787f1270", @@ -935,7 +935,7 @@ "_journal/2024-10/2024-10-31.md": "8c5e70f566953974f252da9472e527f0", "_journal/2024-10/2024-10-30.md": "054bdbf52843fa2445f6b9f91d5ca46e", "_journal/2024-10/2024-10-29.md": "432b3b073dafd54421ff6f7182ab9a58", - "c17/alignment.md": "94c20731411dc244c13bc00bbf924589", + "c17/alignment.md": "bda6444de141aeb8072d3ef2971c4d30", "_journal/2024-11-05.md": "6a599e6bc9dcd12a0940956285ae4d00", "_journal/2024-11-06.md": "c91de8a099cfee2514e1054400996e76", "_journal/2024-11/2024-11-05.md": "79a1304037e18fefa135d576db040784", @@ -946,7 +946,7 @@ "_journal/2024-11/2024-11-08.md": "806bbade5f8339579287687f9433334e", "_journal/2024-11/2024-11-07.md": "434ec3f15d7065ea740127aa8477dd17", "x86-64/directives.md": "019c1c1d04efb26c3e8758aac4543cc7", - "geometry/cartesian.md": "1977d6fd05109456cf1824822b836b02", + "geometry/cartesian.md": "71dd4816e1ead4db46b8189e7d275f34", "geometry/index.md": "cac68c1b624dbb0552e56cce47bcc21d", "_journal/2024-11-10.md": "5478337fd2017b99d0b359713a511e66", "_journal/2024-11/2024-11-09.md": "46f3a640223ef533f4523837b67b57c3", @@ -966,7 +966,7 @@ "_journal/2024-11/2024-11-21.md": "951b6034d60a40dbd8201c50abf0dbb9", "_journal/2024-11/2024-11-20.md": "951b6034d60a40dbd8201c50abf0dbb9", "_journal/2024-11/2024-11-19.md": "d879f57154cb27cb168eb1f1f430e312", - "set/cardinality.md": "9609eca0fe79de40e43d56cd0ef45985", + "set/cardinality.md": "b396048272652dcf49c875e723377ff4", "geometry/area.md": "7f947bb5ac782495a1fb4a63bb2463e7", "_journal/2024-11-23.md": "911f82ab8aede5ecdb96493aef64b0b9", "_journal/2024-11/2024-11-22.md": "51117030e2364dbce3a8d507dead86ae", @@ -999,7 +999,7 @@ "_journal/2024-12/2024-12-04.md": "965f6619edf1002d960203e3e12a413b", "_journal/2024-12-06.md": "d75323d0fec57f4fc1f13cb4370df18d", "_journal/2024-12/2024-12-05.md": "4f3b1e7a43e01cc97b0eed6fbc6c1f96", - "calculus/integrals.md": "a7ef5031ca474cd9d37c1aea85e96237", + "calculus/integrals.md": "e8d36a8fae9cd785374f419996ef5269", "_journal/2024-12-07.md": "bfb6c4db0acbacba19f03a04ec29fa5c", "_journal/2024-12/2024-12-06.md": "d73b611d2d15827186a0252d9b9a6580", "_journal/2024-12-08.md": "5662897539b222db1af45dcd217f0796", @@ -1017,17 +1017,23 @@ "_journal/2024-12/2024-12-11.md": "e7393c01b44c5b804f86f45b8b899b59", "encoding/uri.md": "394abe477b882e1414dd2fb151fea823", "encoding/xml.md": "5b345c09ac60821f88ed5b592e411df6", - "encoding/rdf.md": "b32d6c0a2249794da89d9c7fb7ca325e", + "encoding/rdf.md": "f5c2c0156046a9474bae3577e739c5ce", "_journal/2024-12-15.md": "be66c8808d8bb66d4e7b91db7c93c94a", "_journal/2024-12/2024-12-14.md": "d2223f90fd1ce3d82a4fbb6828a1ec56", "_journal/2024-12/2024-12-13.md": "d2223f90fd1ce3d82a4fbb6828a1ec56", "_journal/2024-12/2024-12-12.md": "59e71caa4e9ebdb11a7c7549c33bed20", "linkers/object-files.md": "77767f310330b8650a5023dd0522226c", - "_journal/2024-12-16.md": "6c03d49c05eb47486aae48aa40dcef9f", + "_journal/2024-12-16.md": "d867a62a955f3d080ae25f31464d53c4", "_journal/2024-12/2024-12-15.md": "be66c8808d8bb66d4e7b91db7c93c94a", - "linkers/elf.md": "83a27a7d71ca26f8f034770a31d58fa0", + "linkers/elf.md": "64262b8ec7d85bab4bd32a238beac638", "c17/strings/printf.md": "8b67cfbccaf35dd9488b73e7e5555405", - "c17/strings/index.md": "3fa6f42967f3cc786740bb8537c62682" + "c17/strings/index.md": "3fa6f42967f3cc786740bb8537c62682", + "_journal/2024-12-17.md": "ae55db66c9835876c4a0343ac0806951", + "_journal/2024-12/2024-12-16.md": "03fe3263baca3bba63a9129595733d5f", + "_journal/2024-12-18.md": "9cf32faceb4d52bf2303e8dcce7bda20", + "_journal/2024-12/2024-12-17.md": "afea61b79ad8dcb02a99e385dce1bbc2", + "_journal/2024-12-19.md": "0064c8a827222092a55b3d896033b84b", + "_journal/2024-12/2024-12-18.md": "06f32ecf5bc088c6db98476fadfd0b5c" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-12-19.md b/notes/_journal/2024-12-19.md new file mode 100644 index 0000000..bc69000 --- /dev/null +++ b/notes/_journal/2024-12-19.md @@ -0,0 +1,12 @@ +--- +title: "2024-12-19" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Additional flashcards on scaling and shifting. +* Notes on C [[alignment#Bit-Fields|bit-fields]]. \ No newline at end of file diff --git a/notes/_journal/2024-12-16.md b/notes/_journal/2024-12/2024-12-16.md similarity index 73% rename from notes/_journal/2024-12-16.md rename to notes/_journal/2024-12/2024-12-16.md index 313979d..b1e67f8 100644 --- a/notes/_journal/2024-12-16.md +++ b/notes/_journal/2024-12/2024-12-16.md @@ -2,11 +2,12 @@ title: "2024-12-16" --- -- [ ] Anki Flashcards +- [x] Anki Flashcards - [x] KoL - [ ] OGS - [ ] Sheet Music (10 min.) - [ ] Korean (Read 1 Story) * Notes on sections found within an [[elf|ELF]] file. -* Additional examples on cardinal numbers and their relation to the recursion theorem. \ No newline at end of file +* Additional examples on cardinal numbers and their relation to the recursion theorem. +* Reification in the context of RDF. \ No newline at end of file diff --git a/notes/_journal/2024-12/2024-12-17.md b/notes/_journal/2024-12/2024-12-17.md new file mode 100644 index 0000000..d825dd5 --- /dev/null +++ b/notes/_journal/2024-12/2024-12-17.md @@ -0,0 +1,11 @@ +--- +title: "2024-12-17" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* More notes on [[elf#Relocatable Object Files|relocatable object files]]. \ No newline at end of file diff --git a/notes/_journal/2024-12/2024-12-18.md b/notes/_journal/2024-12/2024-12-18.md new file mode 100644 index 0000000..8b62f38 --- /dev/null +++ b/notes/_journal/2024-12/2024-12-18.md @@ -0,0 +1,11 @@ +--- +title: "2024-12-18" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Identities regarding cardinal numbers. \ No newline at end of file diff --git a/notes/c17/alignment.md b/notes/c17/alignment.md index f3006ce..f4f7709 100644 --- a/notes/c17/alignment.md +++ b/notes/c17/alignment.md @@ -468,7 +468,482 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% +## Bit-Fields + +A member of a structure or union may be declared to consist of a specified number of bits (including a sign bit, if any). Such a member is called a **bit-field**; its width is preceded by a colon. Its width must be an ICE with a nonnegative value that does not exceed the width of an object of the type that would be specified were the colon and expression omitted. + +A bit-field shall have a type is a qualified or unqualified version of `_Bool`, `signed int`, `unsigned int`, or some other implementation-defined type. + +%%ANKI +Basic +A bit-field can be declared as members of what? +Back: `struct`s or `union`s. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +How is a bit-field declared? +Back: As a member of a `struct` or `union` with width preceded by a colon. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What *kind* of expression must the width of a bit-field be? +Back: An ICE. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What non-implementation-defined types can a bit-field be declared with? +Back: `_Bool`, `signed int`, or `unsigned int`. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What types can a bit-field be declared with? +Back: `_Bool`, `signed int`, `unsigned int`, or some other implementation-defined type. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What is the sign of the following bit-field? +```c +struct foo { int bar : 1; }; +``` +Back: N/A. This is implementation-defined. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What is the sign of the following bit-field? +```c +struct foo { signed bar : 1; }; +``` +Back: Signed. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What is the sign of the following bit-field? +```c +struct foo { unsigned bar : 1; }; +``` +Back: Unsigned. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Which unary operator cannot be applied to a bit-field object? +Back: `&` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What compilation error exists in the following translation unit? +```c +#include + +struct foo { int a; int b : 4; }; + +int main(void) { + struct foo bar = { .a = 1, .b = 1 }; + printf("%p\n", (void *)&bar.b); +} +``` +Back: Cannot take the address-of a bit-field. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What compilation error exists in the following translation unit? +```c +#include + +struct foo { int a; int b : 4; }; + +int main(void) { + struct foo bar = { .a = 1, .b = 1 }; + printf("%p\n", (void *)&bar.a); +} +``` +Back: N/A. This compiles correctly. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What is an unnamed bit-field? +Back: A bit-field with no declarator. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +### Packing + +An implementation may allocate any addressable storage unit large enough to hold a bit-field. If enough space remains, a bit-field that immediately follows another in a structure shall be packed into adjacent bits of the same unit. + +If insufficient space remains, whether a bit-field that does not fit is put into the next unit or overlaps adjacent units is implementation-defined. + +A bit-field structure member with a width of `0` indicates that no further bit-field is to be packed into the unit in which the previous bit-field, if any, was placed. + +%%ANKI +Basic +What are the addressable storage units supported in x86-64? +Back: A byte, word, double word, or quad word. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: x86-64 + +END%% + +%%ANKI +Basic +What does the C standard mean by an "addressable storage unit"? +Back: A memory unit that can be directly addressed/manipulated by the processor. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Does x86-64 allow bit-fields to overlap adjacent addressable storage units? +Back: No. +Reference: Raymond, Eric. “The Lost Art of Structure Packing.” Accessed November 4, 2024. [http://www.catb.org/esr/structure-packing/](http://www.catb.org/esr/structure-packing/). +Tags: x86-64 + +END%% + +%%ANKI +Basic +What does a bit-field of width `0` indicate? +Back: No subsequent bit-field can be packed into the unit in which the previous bit-field, if any, was placed. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What is wrong with the following `struct` definition? +```c +struct foo { unsigned bar : 0; }; +``` +Back: A bit-field of width `0` cannot have a declarator. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +How is the following `struct` correctly written? +```c +struct foo { unsigned bar : 0; }; +``` +Back: +```c +struct foo { unsigned : 0; }; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Assume a `4`-byte `unsigned int`. What is wrong with the following `struct` definition? +```c +struct foo { unsigned bar : 31; }; +``` +Back: N/A. This is correct. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). +Tags: x86-64 + +END%% + +%%ANKI +Basic +Assume a `4`-byte `unsigned int`. What is wrong with the following `struct` definition? +```c +struct foo { unsigned bar : 32; }; +``` +Back: N/A. This is correct. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). +Tags: x86-64 + +END%% + +%%ANKI +Basic +Assume a `4`-byte `unsigned int`. What is wrong with the following `struct` definition? +```c +struct foo { unsigned bar : 33; }; +``` +Back: The width of a bit-field cannot exceed its types (in this case `unsigned int`). +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). +Tags: x86-64 + +END%% + +%%ANKI +Basic +Assume a byte-sized storage unit, no overlapping units, and low-to-high order. How is the following packed in memory? +```c +struct foo { + signed a : 4; + signed b : 2; + signed c : 4; +}; +``` +Back: +```c +struct foo { + _padding : 2; // 2 bits + signed b : 2; // 2 bits + signed a : 4; // 4 bits + _padding : 4; // 4 bits + signed c : 4; // 4 bits +}; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Assume a byte-sized storage unit, overlapping units, and low-to-high order. How is the following packed in memory? +```c +struct foo { + signed a : 4; + signed : 0; + signed c : 4; +}; +``` +Back: +```c +struct foo { + _padding : 4; // 4 bits + signed a : 4; // 4 bits + _padding : 4; // 4 bits + signed c : 4; // 4 bits +}; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Assume a byte-sized storage unit, overlapping units, and high-to-low order. How is the following packed in memory? +```c +struct foo { + signed a : 4; + signed : 0; + signed c : 4; +}; +``` +Back: +```c +struct foo { + signed a : 4; // 4 bits + _padding : 4; // 4 bits + signed c : 4; // 4 bits + _padding : 4; // 4 bits +}; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Assume a `32`-bit storage unit, no overlapping units, and low-to-high order. How is the following packed in memory? +```c +struct foo { + signed a : 4; + signed b : 2; + signed c : 4; +}; +``` +Back: +```c +struct foo { + _padding : 22; // 22 bits + signed c : 4; // 4 bits + signed b : 2; // 2 bits + signed a : 4; // 4 bits +}; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Assume a byte-sized storage unit, overlapping units, and low-to-high order. How is the following packed in memory? +```c +struct foo { + signed a : 4; + signed b : 2; + signed c : 4; +}; +``` +Back: +```c +struct foo { + _padding : 6; // 6 bits + signed c : 4; // 4 bits + signed b : 2; // 2 bits + signed a : 4; // 4 bits +}; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Assume a `32`-bit storage unit, no overlapping units, and low-to-high order. How is the following packed in memory? +```c +struct foo { + signed a : 4; + signed : 0; + signed c : 4; +}; +``` +Back: +```c +struct foo { + _padding : 28; // 28 bits + signed c : 4; // 4 bits + _padding : 28; // 28 bits + signed a : 4; // 4 bits +}; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Assume a byte-sized storage unit, overlapping units, and high-to-low order. How is the following packed in memory? +```c +struct foo { + signed a : 4; + signed b : 2; + signed c : 4; +}; +``` +Back: +```c +struct foo { + signed a : 4; // 4 bits + signed b : 2; // 2 bits + signed c : 4; // 4 bits + _padding : 6; // 6 bits +}; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Assume a byte-sized storage unit, no overlapping units, and high-to-low order. How is the following packed in memory? +```c +struct foo { + signed a : 4; + signed b : 2; + signed c : 4; +}; +``` +Back: +```c +struct foo { + signed a : 4; // 4 bits + signed b : 2; // 2 bits + _padding : 2; // 2 bits + signed c : 4; // 4 bits + _padding : 4; // 4 bits +}; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +Assume a `32`-bit storage unit, overlapping units, and high-to-low order. How is the following packed in memory? +```c +struct foo { + signed a : 4; + signed b : 2; + signed c : 4; +}; +``` +Back: +```c +struct foo { + signed a : 4; // 4 bits + signed b : 2; // 2 bits + signed c : 4; // 4 bits + _padding : 22; // 22 bits +}; +``` +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What implementation-defined property guarantees tightly-packed bit-fields? +Back: Whether bit-fields can overlap adjacent addressable storage units. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Cloze +An implementation may allocate any {addressable storage} unit large enough to hold a bit-field. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What does it mean for bit-fields to be allocated low-to-high? +Back: They are packed starting from the least significant bit of the addressable storage unit. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + +%%ANKI +Basic +What does it mean for bit-fields to be allocated high-to-low? +Back: They are packed starting from the most significant bit of the addressable storage unit. +Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). + +END%% + ## Bibliography * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +* “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf). * Raymond, Eric. “The Lost Art of Structure Packing.” Accessed November 4, 2024. [http://www.catb.org/esr/structure-packing/](http://www.catb.org/esr/structure-packing/). \ No newline at end of file diff --git a/notes/c17/index.md b/notes/c17/index.md index b4204c2..540b398 100644 --- a/notes/c17/index.md +++ b/notes/c17/index.md @@ -645,7 +645,7 @@ END%% %%ANKI Basic -Which component of a compiler driver produces ASCII relocatable object files? +Which component of a compiler driver produces relocatable object files? Back: The assembler. Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. @@ -655,7 +655,7 @@ The operating system provides a function called the **loader** that copies the c %%ANKI Basic -What function does the OS provider to run an executable object file? +What function does the OS provide to run executable object files? Back: The loader. Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. diff --git a/notes/calculus/integrals.md b/notes/calculus/integrals.md index 5fcb11a..d369cd1 100644 --- a/notes/calculus/integrals.md +++ b/notes/calculus/integrals.md @@ -7,7 +7,7 @@ tags: - integral --- -## Overview +## Step Functions Let $s$ be a step function defined on [[intervals|interval]] $[a, b]$, and let $P = \{x_0, x_1, \ldots, x_n\}$ be a [[intervals#Partitions|partition]] of $[a, b]$ such that $s$ is constant on the open subintervals of $P$. Denote by $s_k$ the constant value that $s$ takes in the $k$th open subinterval, so that $$s(x) = s_k \quad\text{if}\quad x_{k-1} < x < x_k, \quad k = 1, 2, \ldots, n.$$ diff --git a/notes/encoding/binary.md b/notes/encoding/binary.md index bf557a7..d17c554 100644 --- a/notes/encoding/binary.md +++ b/notes/encoding/binary.md @@ -1,6 +1,6 @@ --- title: Binary -TARGET DECK: Obsidian::H&SS +TARGET DECK: Obsidian::STEM FILE TAGS: binary tags: - binary @@ -176,6 +176,42 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% +%%ANKI +Cloze +A big-endian system stores the {most} significant byte at the {smallest} memory address. +Reference: “Endianness,” in _Wikipedia_, December 19, 2024, [https://en.wikipedia.org/w/index.php?title=Endianness](https://en.wikipedia.org/w/index.php?title=Endianness&oldid=1264018894). + +END%% + +%%ANKI +Cloze +A big-endian system stores the {least} significant byte at the {largest} memory address. +Reference: “Endianness,” in _Wikipedia_, December 19, 2024, [https://en.wikipedia.org/w/index.php?title=Endianness](https://en.wikipedia.org/w/index.php?title=Endianness&oldid=1264018894). + +END%% + +%%ANKI +Cloze +A little-endian system stores the {least} significant byte at the {smallest} memory address. +Reference: “Endianness,” in _Wikipedia_, December 19, 2024, [https://en.wikipedia.org/w/index.php?title=Endianness](https://en.wikipedia.org/w/index.php?title=Endianness&oldid=1264018894). + +END%% + +%%ANKI +Cloze +A little-endian system stores the {most} significant byte at the {largest} memory address. +Reference: “Endianness,” in _Wikipedia_, December 19, 2024, [https://en.wikipedia.org/w/index.php?title=Endianness](https://en.wikipedia.org/w/index.php?title=Endianness&oldid=1264018894). + +END%% + +%%ANKI +Basic +Which of little- or big-endian more closely mirrors the way numbers are written in English? +Back: Big-endian. +Reference: “Endianness,” in _Wikipedia_, December 19, 2024, [https://en.wikipedia.org/w/index.php?title=Endianness](https://en.wikipedia.org/w/index.php?title=Endianness&oldid=1264018894). + +END%% + ```c #include #include @@ -192,4 +228,5 @@ The above snippet can be used to check endianness on the current machine. If big ## Bibliography -* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. \ No newline at end of file +* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +* “Endianness,” in _Wikipedia_, December 19, 2024, [https://en.wikipedia.org/w/index.php?title=Endianness](https://en.wikipedia.org/w/index.php?title=Endianness&oldid=1264018894). \ No newline at end of file diff --git a/notes/encoding/floating-point.md b/notes/encoding/floating-point.md index 5403451..586df37 100644 --- a/notes/encoding/floating-point.md +++ b/notes/encoding/floating-point.md @@ -442,7 +442,7 @@ END%% %%ANKI Cloze -The exponent field of a `float` has {`8`} bits and a `double` has {`11`} bits. +The exponent field of a `float` has {`8`} bits whereas a `double` has {`11`} bits. Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c17 diff --git a/notes/encoding/rdf.md b/notes/encoding/rdf.md index f11f9f6..db528b3 100644 --- a/notes/encoding/rdf.md +++ b/notes/encoding/rdf.md @@ -128,6 +128,114 @@ Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web END%% +## Reification + +**Reification** refers to the process by which an abstract idea is made concrete. In the context of RDF, it refers to writing RDF statements about RDF statements. + +%%ANKI +Basic +What is reification? +Back: The process by which an abstract idea is made concrete. +Reference: “Reification (Knowledge Representation),” in _Wikipedia_, October 3, 2023, [https://en.wikipedia.org/w/index.php?title=Reification_(knowledge_representation)](https://en.wikipedia.org/w/index.php?title=Reification_(knowledge_representation)&oldid=1178437461). + +END%% + +%%ANKI +Basic +In the context of RDF, what does reification typically refer to? +Back: Creating RDF statements about RDF statements. +Reference: “Reification (Knowledge Representation),” in _Wikipedia_, October 3, 2023, [https://en.wikipedia.org/w/index.php?title=Reification_(knowledge_representation)](https://en.wikipedia.org/w/index.php?title=Reification_(knowledge_representation)&oldid=1178437461). + +END%% + +%%ANKI +Basic +In the context of RDF, what abstract notion is typically reified? +Back: RDF statements. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What term describes making an abstract concept concrete? +Back: Reification. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +This kind of metadata about statements often take the forms of: + +* **Provenance**. Information about the source of a statement. +* **Likelihood**. Information quantifying some probability regarding the statement. +* **Context**. Information about a setting in which a statement holds. +* **Time frame**. Information that holds within a particular period of time. + +%%ANKI +Basic +What might reifying a statement with respect to its provenance refer to? +Back: Information about the source of the statement. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What "form" of reification does the following sentence take on? $$\text{Wikipedia says Shakespeare wrote Hamlet.}$$ +Back: Provenance. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What might reifying a statement with respect to its likelihood refer to? +Back: Information quantifying some probability of the statement holding. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What "form" of reification does the following sentence take on? $$\text{It is 90\% probable that Shakespeare wrote Hamlet.}$$ +Back: Likelihood. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What might reifying a statement with respect to its context refer to? +Back: Information about a setting in which the statement holds. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What "form" of reification does the following sentence take on? $$\text{Kenneth Branagh played Hamlet in the movie.}$$ +Back: Context. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What might reifying a statement with respect to its time frame refer to? +Back: Information about a period of time in which the statement holds. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What "form" of reification does the following sentence take on? $$\text{Hamlet plays on Broadway Jan. 11th through Mar. 12th.}$$ +Back: Time frame. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + ## Bibliography -* Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. \ No newline at end of file +* Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. +* “Reification (Knowledge Representation),” in _Wikipedia_, October 3, 2023, [https://en.wikipedia.org/w/index.php?title=Reification_(knowledge_representation)](https://en.wikipedia.org/w/index.php?title=Reification_(knowledge_representation)&oldid=1178437461). \ No newline at end of file diff --git a/notes/geometry/cartesian.md b/notes/geometry/cartesian.md index c24d1e9..47d30e4 100644 --- a/notes/geometry/cartesian.md +++ b/notes/geometry/cartesian.md @@ -131,13 +131,13 @@ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Int END%% -## Translations +## Transformations -There are two kinds of translations that we can do to a graph: **shifting** and **scaling**. A **reflection** is a special case of scaling. +There are two kinds of transformations that we can do to a graph: **shifting** and **scaling**. A **reflection** is a special case of scaling. %%ANKI Basic -What are the two kinds of translations that can be done to a graph? +What are the two kinds of transformations that can be done to a graph? Back: Shifting and scaling. Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). @@ -145,7 +145,7 @@ END%% %%ANKI Basic -Which of the two kinds of translations is reflection a special case of? +Which of the two kinds of transformations is reflection a special case of? Back: Scaling. Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). @@ -165,6 +165,150 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces END%% +%%ANKI +Basic +Given function $f$, which of $a$, $b$, $c$, and/or $d$ relate to shifting? $$y = af(b[x - c]) + d$$ +Back: $c$ and $d$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Given function $f$, which of $a$, $b$, $c$, and/or $d$ relate to a horizontal transformation? $$y = af(b[x - c]) + d$$ +Back: $b$ and $c$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Given function $f$, which of $a$, $b$, $c$, and/or $d$ relate to a vertical transformation? $$y = af(b[x - c]) + d$$ +Back: $a$ and $d$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Given function $f$, which of $a$, $b$, $c$, and/or $d$ relate to scaling? $$y = af(b[x - c]) + d$$ +Back: $a$ and $b$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Given function $f$, which of $a$, $b$, $c$, and/or $d$ apply a transformation that "acts normally"? $$y = af(b[x - c]) + d$$ +Back: $a$ and $d$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Given function $f$, which of $a$, $b$, $c$, and/or $d$ apply a transformation that "acts inversely"? $$y = af(b[x - c]) + d$$ +Back: $b$ and $c$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Which of horizontal and/or vertical transformations "act inversely"? +Back: Horizontal transformations. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Which of horizontal and/or vertical transformations "act normally"? +Back: Vertical transformations. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +How is the graph of $y = f(x)$ transformed in the graph of $y = \frac{1}{2}f(\frac{x}{3})$? +Back: It's horizontally scaled by $3$ and vertically compressed by $2$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Consider $y = f(x)$. Is $y = af(bx)$ first scaled vertically or horizontally? +Back: N/A. There is no order of operations in this case. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +How is the graph of $y = f(x)$ transformed in the graph of $y = f(x - 3) + 2$? +Back: It's shifted right by $3$ and shifted up by $2$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Consider $y = f(x)$. Is $y = f(x + a) + b$ first shifted vertically or horizontally? +Back: N/A. There is no order of operations in this case. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +How is the graph of $y = f(x)$ transformed in the graph of $y = 2f(x) + 5$? +Back: It's vertically scaled by $2$ and then shifted up by $5$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Consider $y = f(x)$. Is $y = af(x) + b$ first scaled or shifted? +Back: Scaled. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Consider $y = f(x)$. Is $y = f(ax) + b$ first scaled or shifted? +Back: Scaled. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +How is the graph of $y = f(x)$ transformed in the graph of $y = f(2x - 3)$? +Back: It's horizontally compressed by $2$ and then shifted right by $3/2$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $2f(x - 3) - 5$? +Back: $(1, 8)$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has range $[4, 8]$. What is the range of $2f(x - 3) - 5$? +Back: $[3, 11]$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + ### Shifting A **vertical shift** adds/subtracts a constant to every $y$-coordinate of a graph, leaving the $x$-coordinate unchanged. A **horizontal shift** adds/subtracts a constant to every $x$-coordinate of a graph, leaving the $y$-coordinate unchanged. @@ -179,7 +323,7 @@ END%% %%ANKI Basic -Which of the two fundamental graph translations is considered "rigid"? +Which of the two fundamental graph transformations is considered "rigid"? Back: Shifts. Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). @@ -215,7 +359,7 @@ END%% %%ANKI Basic -Let $f(x)$ be a function and $k$ be a constant. What kind of translation is $f(x + k)$? +Let $f(x)$ be a function and $k$ be a constant. What kind of transformations is $f(x + k)$? Back: A horizontal shift. Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). @@ -223,7 +367,7 @@ END%% %%ANKI Basic -Let $f(x)$ be a function and $k$ be a constant. What kind of translation is $f(x) + k$? +Let $f(x)$ be a function and $k$ be a constant. What kind of transformations is $f(x) + k$? Back: A vertical shift. Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). @@ -311,6 +455,70 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces END%% +%%ANKI +Basic +Horizontal shifting of a function affects which of its range and/or domain? +Back: The domain. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(x - 2)$? +Back: $(0, 7)$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(x + 2)$? +Back: $(-4, 3)$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(x) + 2$? +Back: $(-2, 5)$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Vertical shifting of a function affects which of its range and/or domain? +Back: The range. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(x - 2)$? +Back: $[4, 8]$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(x) - 2$? +Back: $[2, 6]$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(x) + 2$? +Back: $[6, 10]$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + ### Scaling A **vertical scaling** will multiply/divide every $y$-coordinate of a graph, leaving the $x$-coordinate unchanged. A **horizontal scaling** will multiply/divide every $x$-coordinate of a graph, leaving the $y$-coordinate unchanged. @@ -341,7 +549,7 @@ END%% %%ANKI Basic -Let $f(x)$ be a function and $k$ be a constant. What kind of translation is $kf(x)$? +Let $f(x)$ be a function and $k$ be a constant. What kind of transformation is $kf(x)$? Back: A vertical scaling. Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). @@ -349,7 +557,7 @@ END%% %%ANKI Basic -Let $f(x)$ be a function and $k$ be a constant. What kind of translation is $f(kx)$? +Let $f(x)$ be a function and $k$ be a constant. What kind of transformation is $f(kx)$? Back: A horizontal scaling. Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). @@ -425,6 +633,144 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces END%% +%%ANKI +Basic +Horizontal scaling of a function affects which of its range and/or domain? +Back: The domain. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(2x)$? +Back: $(-1, 5/2)$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $2f(x)$? +Back: $(-2, 5)$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(x / 2)$? +Back: $(-4, 10)$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Vertical scaling of a function affects which of its range and/or domain? +Back: The range. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(2x)$? +Back: $[4, 8]$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has range $[4, 8]$. What is the range of $2f(x)$? +Back: $[8, 16]$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(x) / 2$? +Back: $[2, 4]$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +### Reflecting + +A **reflection** is a special case of a [[#Scaling|scaling]]. + +%%ANKI +Basic +A reflection is a special case of what other kind of transformation? +Back: A scaling. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Cloze +To reflect a function's graph about the {$y$}-axis, multiply its {inputs} by $-1$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Cloze +To reflect a function's graph about the {$x$}-axis, multiply its {outputs} by $-1$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Consider the graph of $f(x)$. Is $f(-x)$ its horizontal or vertical reflection? +Back: Horizontal (reflection along the $y$-axis). +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(-x)$? +Back: $(-5, 2)$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has domain $(-2, 5)$. What is the range of $f(-x)$? +Back: The same as that of $f(x)$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Consider the graph of $f(x)$. Is $-f(x)$ its horizontal or vertical reflection? +Back: Vertical (reflection along the $x$-axis). +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has range $[4, 8]$. What is the domain of $-f(x)$? +Back: The same as that of $f(x)$. +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + +%%ANKI +Basic +Suppose $f(x)$ has range $[4, 8]$. What is the range of $-f(x)$? +Back: $[-8, -4]$ +Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). + +END%% + ## Bibliography * “Cartesian Coordinate System,” in _Wikipedia_, October 21, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system](https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system&oldid=1252434514). diff --git a/notes/linkers/elf.md b/notes/linkers/elf.md index 399e911..acf2385 100644 --- a/notes/linkers/elf.md +++ b/notes/linkers/elf.md @@ -12,8 +12,6 @@ tags: Modern [[x86-64/index|x86-64]] Linux systems use the ELF (Executable and Linkable Format) object file format. -![[elf.png]] - %%ANKI Basic What object file format do modern x86-64 Linux machines typically use? @@ -38,11 +36,46 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% -### Relocatable Object Files +## Relocatable Object Files -A typical ELF relocatable object file contains the following sections: +![[elf.png]] -#### `.text` +%%ANKI +Basic +A relocatable object file is typically broken up into what three regions? +Back: The header, sections, and the section header table. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +In a relocatable object file, what exists between the header and section header table? +Back: The sections. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Cloze +A relocatable object file consists of a {header}, {sections}, and a {section header table}, in that order. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +Where in a relocatable object file does the section header table exist? +Back: At the end. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +## Sections + +A typical ELF object file contains the following sections: + +### `.text` The machine code of the compiled program. @@ -69,7 +102,7 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% -#### `.rodata` +### `.rodata` Read-only data such as the format strings in [[c17/strings/printf|printf]] statements and [[conditions#JMP|jump tables]] for switch statements. @@ -114,7 +147,7 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% -#### `.data` +### `.data` Global and static C variables initialized to a non-zero value. @@ -187,7 +220,7 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% -#### `.bss` +### `.bss` Uninitialized global and static C variables, along with any global or static variables initialized to zero. @@ -298,7 +331,7 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% -#### `.symtab` +### `.symtab` A symbol table with information about functions and global variables defined and referenced in the program. @@ -326,6 +359,96 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% +### `.rel.text` + +A list of locations in the `.text` section that will need to be modified when the linker combines this object file with others. + +%%ANKI +Basic +What does the `.rel.text` section of an ELF file contain? +Back: Relocation entries for the `.text` section. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +Why is the `.rel.text` ELF section named the way it is? +Back: It's short for **rel**ocation entries for the `.text` section. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Cloze +The {`.rel.text`} section contains {relocation entries} for the `.text` section. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +### `.rel.data` + +A list of locations in the `.data` section that will need to be modified when the linker combines this object file with others. + +%%ANKI +Basic +What does the `.rel.data` section of an ELF file contain? +Back: Relocation entries for the `.data` section. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +Why is the `.rel.data` ELF section named the way it is? +Back: It's short for **rel**ocation entries for the `.data` section. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Cloze +The {`.rel.data`} section contains {relocation entries} for the `.data` section. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +### `.strtab` + +A string table for the symbol tables in the `.symtab` section as well as for section names in the section headers. It is a sequence of `NUL`-terminated character strings. + +%%ANKI +Basic +What does the `.strtab` section of an ELF file contain? +Back: A string table. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +Why is the `.strtab` ELF section named the way it is? +Back: It is short for **str**ing **tab**le. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +The `.strtab` provides information for what other, non-debug ELF section? +Back: `.symtab` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +Section names are contained in what ELF section? +Back: `.strtab` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + ## Bibliography * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. \ No newline at end of file diff --git a/notes/set/cardinality.md b/notes/set/cardinality.md index 03b1dab..58526f2 100644 --- a/notes/set/cardinality.md +++ b/notes/set/cardinality.md @@ -552,7 +552,7 @@ END%% %%ANKI Basic Suppose sets $A$ and $B$ are finite. When is $A \cup B$ infinite? -Back: The union of two finite sets is always finite. +Back: N/A. The union of two finite sets is always finite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% @@ -591,8 +591,8 @@ END%% %%ANKI Basic -Suppose sets $A$ and $B$ are finite. When is $A \cap B$ finite? -Back: The intersection of two finite sets is always finite. +Suppose sets $A$ and $B$ are finite. When is $A \cap B$ infinite? +Back: N/A. The intersection of two finite sets is always finite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% @@ -615,20 +615,12 @@ END%% %%ANKI Basic -Suppose sets $A$ and $B$ are finite. When is $A \times B$ finite? -Back: The Cartesian product of two finite sets is always finite. +Suppose sets $A$ and $B$ are finite. When is $A \times B$ infinite? +Back: N/A. The Cartesian product of two finite sets is always finite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% -%%ANKI -Basic -Let $m, n \in \omega$ such that $A \approx m$ and $B \approx n$. What does $\mathop{\text{card}}(A \times B)$ evaluate to? -Back: $m \cdot n$ -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - ### Addition Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa + \lambda = \mathop{\text{card}}(K \cup L)$, where $K$ and $L$ are any disjoint sets of cardinality $\kappa$ and $\lambda$, respectively. @@ -641,14 +633,6 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% -%%ANKI -Basic -Let $K$ and $L$ be sets. What does $\mathop{\text{card}}(K \cup L)$ evaluate to? -Back: N/A. $K$ and $L$ must be disjoint sets for evaluation to make sense. -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - %%ANKI Basic Let $K$ and $L$ be disjoint sets. What does $\mathop{\text{card}}(K \cup L)$ evaluate to? @@ -659,7 +643,7 @@ END%% %%ANKI Basic -Let $K \approx \kappa$ and $L \approx \lambda$. What is necessary for $\mathop{\text{card}}(K \cup L) \approx \kappa + \lambda$? +Let $\mathop{\text{card}}(K) = \kappa$ and $\mathop{\text{card}}(L) = \lambda$. What is necessary for $\mathop{\text{card}}(K \cup L) = \kappa + \lambda$? Back: That $K$ and $L$ are disjoint. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). @@ -696,6 +680,62 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +What cardinal number does $0 + \aleph_0$ evaluate to? +Back: $\aleph_0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Expression $0 + \aleph_0$ corresponds to the cardinality of what set? +Back: $\varnothing \cup \omega$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $n \in \omega$. What cardinal number does $n^+ + \aleph_0$ evaluate to? +Back: $\aleph_0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $n \in \omega$. Expression $n + \aleph_0$ corresponds to the cardinality of what set? +Back: $(n \times \{0\}) \cup (\omega \times \{1\})$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What cardinal number does $\aleph_0 + \aleph_0$ evaluate to? +Back: $\aleph_0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Expression $\aleph_0 + \aleph_0$ corresponds to the cardinality of what set? +Back: $(\omega \times \{0\}) \cup (\omega \times \{1\})$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $\kappa$ be a cardinal number. What cardinal number does $\kappa + 0$ evaluate to? +Back: $0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ### Multiplication Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa \cdot \lambda = \mathop{\text{card}}(K \times L)$, where $K$ and $L$ are any sets of cardinality $\kappa$ and $\lambda$, respectively. @@ -731,6 +771,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +Let $m, n \in \omega$ such that $A \approx m$ and $B \approx n$. What does $\mathop{\text{card}}(A \times B)$ evaluate to? +Back: $m \cdot n$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + %%ANKI Basic How do we prove $2 \cdot 2 = 4$ using the recursion theorem? @@ -755,6 +803,70 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +What cardinal number does $0 \cdot \aleph_0$ evaluate to? +Back: $0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Expression $0 \cdot \aleph_0$ corresponds to the cardinality of what set? +Back: $\varnothing \times \omega$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $n \in \omega$. What cardinal number does $n^+ \cdot \aleph_0$ evaluate to? +Back: $\aleph_0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $n \in \omega$. Expression $n \cdot \aleph_0$ corresponds to the cardinality of what set? +Back: $n \times \omega$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What cardinal number does $\aleph_0 \cdot \aleph_0$ evaluate to? +Back: $\aleph_0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Expression $\aleph_0 \cdot \aleph_0$ corresponds to the cardinality of what set? +Back: $\omega \times \omega$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $\kappa$ be a cardinal number. What cardinal number does $\kappa \cdot 0$ evaluate to? +Back: $0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $\kappa$ be a cardinal number. What cardinal number does $\kappa \cdot 1$ evaluate to? +Back: $\kappa$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ### Exponentiation Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa^\lambda = \mathop{\text{card}}(^LK)$, where $K$ and $L$ are any sets of cardinality $\kappa$ and $\lambda$, respectively. @@ -822,6 +934,62 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre END%% +%%ANKI +Basic +What cardinal number does $0^{\aleph_0}$ evaluate to? +Back: $0$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Expression $0^{\aleph_0}$ corresponds to the cardinality of what set? +Back: $^\omega \varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $\kappa$ be a nonzero cardinal number. What cardinal number does $0^\kappa$ evaluate to? +Back: $0$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What cardinal number does $0^0$ evaluate to? +Back: $1$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $\kappa$ be a cardinal number. Expression $0^\kappa$ corresponds to the cardinality of what set? +Back: $^K\varnothing$ where $\mathop{\text{card}} K = \kappa$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $\kappa$ be a cardinal number. What cardinal number does $\kappa^0$ evaluate to? +Back: $1$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Let $\kappa$ be a cardinal number. Expression $\kappa^0$ corresponds to the cardinality of what set? +Back: $^\varnothing K$ where $\mathop{\text{card}} K = \kappa$.. +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