Graph transformations and C bit-fields.

2024-12-19 20:26:57 -07:00 · 2024-12-19 20:26:57 -07:00 · 5bb40f0018
parent d10dc3d24b
commit 5bb40f0018
14 changed files with 1358 additions and 60 deletions
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    "Basic": [
--- a/notes/_journal/2024-12-19.md
+++ 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]].
--- a/notes/_journal/2024-12/2024-12-16.md
+++ b/notes/_journal/2024-12/2024-12-16.md
@ -2,7 +2,7 @@
 title: "2024-12-16"
 ---

- [ ] Anki Flashcards
+- [x] Anki Flashcards
 - [x] KoL
 - [ ] OGS
 - [ ] Sheet Music (10 min.)
@ -10,3 +10,4 @@ title: "2024-12-16"

 * Notes on sections found within an [[elf|ELF]] file.
 * Additional examples on cardinal numbers and their relation to the recursion theorem.
+* Reification in the context of RDF.
--- a/notes/_journal/2024-12/2024-12-17.md
+++ 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]].
--- a/notes/_journal/2024-12/2024-12-18.md
+++ 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.
--- 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
 <!--ID: 1731165663242-->
 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).
+<!--ID: 1734662614053-->
+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).
+<!--ID: 1734662614057-->
+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).
+<!--ID: 1734662614060-->
+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).
+<!--ID: 1734662614063-->
+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).
+<!--ID: 1734662614068-->
+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).
+<!--ID: 1734662614071-->
+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).
+<!--ID: 1734662614075-->
+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).
+<!--ID: 1734662614079-->
+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).
+<!--ID: 1734662614084-->
+END%%
+
+%%ANKI
+Basic
+What compilation error exists in the following translation unit?
+```c
+#include <stdio.h>
+
+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).
+<!--ID: 1734662614089-->
+END%%
+
+%%ANKI
+Basic
+What compilation error exists in the following translation unit?
+```c
+#include <stdio.h>
+
+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).
+<!--ID: 1734662614094-->
+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).
+<!--ID: 1734662614099-->
+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
+<!--ID: 1734662614105-->
+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).
+<!--ID: 1734662614110-->
+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
+<!--ID: 1734662614115-->
+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).
+<!--ID: 1734662614122-->
+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).
+<!--ID: 1734662614129-->
+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).
+<!--ID: 1734662614138-->
+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
+<!--ID: 1734662614146-->
+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
+<!--ID: 1734662614152-->
+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
+<!--ID: 1734662614158-->
+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).
+<!--ID: 1734662614163-->
+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).
+<!--ID: 1734664080915-->
+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).
+<!--ID: 1734664080919-->
+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).
+<!--ID: 1734662614168-->
+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).
+<!--ID: 1734662614174-->
+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).
+<!--ID: 1734664080924-->
+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).
+<!--ID: 1734662614181-->
+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).
+<!--ID: 1734662614188-->
+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).
+<!--ID: 1734662614195-->
+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).
+<!--ID: 1734662614202-->
+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).
+<!--ID: 1734662614208-->
+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).
+<!--ID: 1734664080929-->
+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).
+<!--ID: 1734664080934-->
+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/).
--- 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.
 <!--ID: 1733607099587-->
@ -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.
 <!--ID: 1733607099593-->
--- 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 $$s(x) = s_k \quad\text{if}\quad x_{k-1} < x < x_k, \quad k = 1, 2, \ldots, n.$$

--- 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
 <!--ID: 1707661283782-->
 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).
+<!--ID: 1734664080894-->
+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).
+<!--ID: 1734664080900-->
+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).
+<!--ID: 1734664080903-->
+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).
+<!--ID: 1734664080907-->
+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).
+<!--ID: 1734664080911-->
+END%%
+
 ```c
 #include <stdint.h>
 #include <stdio.h>
@ -193,3 +229,4 @@ 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.
+* “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).
--- 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
 <!--ID: 1710556915030-->
--- 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
 <!--ID: 1722191359882-->
 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).
+<!--ID: 1734385502410-->
+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).
+<!--ID: 1734385502419-->
+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.
+<!--ID: 1734385502422-->
+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.
+<!--ID: 1734385502427-->
+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.
+<!--ID: 1734385502431-->
+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.
+<!--ID: 1734385502435-->
+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.
+<!--ID: 1734385502439-->
+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.
+<!--ID: 1734385502443-->
+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.
+<!--ID: 1734385502448-->
+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.
+<!--ID: 1734385502453-->
+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.
+<!--ID: 1734385502458-->
+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.
+<!--ID: 1734385502463-->
+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.
+* “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).
--- 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
 <!--ID: 1731185808252-->
 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).
 <!--ID: 1733520214987-->
@ -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).
 <!--ID: 1733520269110-->
@ -165,6 +165,150 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces
 <!--ID: 1733522558480-->
 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).
+<!--ID: 1734620943896-->
+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).
+<!--ID: 1734620943904-->
+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).
+<!--ID: 1734620943910-->
+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).
+<!--ID: 1734620943916-->
+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).
+<!--ID: 1734620943921-->
+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).
+<!--ID: 1734620943926-->
+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).
+<!--ID: 1734620943931-->
+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).
+<!--ID: 1734620943937-->
+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).
+<!--ID: 1734656174535-->
+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).
+<!--ID: 1734656174588-->
+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).
+<!--ID: 1734656174596-->
+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).
+<!--ID: 1734656174604-->
+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).
+<!--ID: 1734656174610-->
+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).
+<!--ID: 1734656174615-->
+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).
+<!--ID: 1734656174619-->
+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).
+<!--ID: 1734656174624-->
+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).
+<!--ID: 1734657098882-->
+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).
+<!--ID: 1734657098887-->
+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).
 <!--ID: 1733522558483-->
@ -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).
 <!--ID: 1733520215013-->
@ -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).
 <!--ID: 1733520215016-->
@ -311,6 +455,70 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces
 <!--ID: 1733520215056-->
 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).
+<!--ID: 1734656938097-->
+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).
+<!--ID: 1734656938102-->
+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).
+<!--ID: 1734656938105-->
+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).
+<!--ID: 1734656938109-->
+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).
+<!--ID: 1734656938112-->
+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).
+<!--ID: 1734656938116-->
+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).
+<!--ID: 1734656938120-->
+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).
+<!--ID: 1734656938124-->
+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).
 <!--ID: 1733522558503-->
@ -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).
 <!--ID: 1733522558507-->
@ -425,6 +633,144 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces
 <!--ID: 1733522558555-->
 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).
+<!--ID: 1734656938129-->
+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).
+<!--ID: 1734656938133-->
+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).
+<!--ID: 1734656938137-->
+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).
+<!--ID: 1734656938141-->
+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).
+<!--ID: 1734656938145-->
+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).
+<!--ID: 1734656938149-->
+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).
+<!--ID: 1734656938152-->
+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).
+<!--ID: 1734656938156-->
+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).
+<!--ID: 1734620943942-->
+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).
+<!--ID: 1734620943947-->
+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).
+<!--ID: 1734620943953-->
+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).
+<!--ID: 1734656938160-->
+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).
+<!--ID: 1734656938163-->
+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).
+<!--ID: 1734656938167-->
+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).
+<!--ID: 1734656938171-->
+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).
+<!--ID: 1734656938174-->
+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).
+<!--ID: 1734656938178-->
+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).
--- 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
 <!--ID: 1734356868446-->
 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.
+<!--ID: 1734456879220-->
+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.
+<!--ID: 1734456879305-->
+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.
+<!--ID: 1734456879322-->
+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.
+<!--ID: 1734456879328-->
+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
 <!--ID: 1734367304872-->
 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
 <!--ID: 1734367304878-->
 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
 <!--ID: 1734369188110-->
 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
 <!--ID: 1734369188106-->
 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
 <!--ID: 1734370180083-->
 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.
+<!--ID: 1734456879344-->
+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.
+<!--ID: 1734456879386-->
+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.
+<!--ID: 1734456879393-->
+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.
+<!--ID: 1734456879404-->
+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.
+<!--ID: 1734456879407-->
+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.
+<!--ID: 1734456879414-->
+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.
+<!--ID: 1734456879418-->
+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.
+<!--ID: 1734456879422-->
+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.
+<!--ID: 1734456879427-->
+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.
+<!--ID: 1734456879433-->
+END%%
+
 ## Bibliography

 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
--- 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).
 <!--ID: 1733675522739-->
 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).
 <!--ID: 1733675522748-->
 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).
 <!--ID: 1733675522751-->
 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).
-<!--ID: 1733693785295-->
-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
 <!--ID: 1733710439132-->
 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).
-<!--ID: 1733710439139-->
-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).
 <!--ID: 1733710439146-->
@ -696,6 +680,62 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1734374219325-->
 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).
+<!--ID: 1734520487310-->
+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).
+<!--ID: 1734520487313-->
+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).
+<!--ID: 1734520487316-->
+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).
+<!--ID: 1734520487319-->
+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).
+<!--ID: 1734520487326-->
+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).
+<!--ID: 1734520487330-->
+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).
+<!--ID: 1734520487333-->
+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
 <!--ID: 1733710439162-->
 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).
+<!--ID: 1733693785295-->
+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
 <!--ID: 1734374219328-->
 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).
+<!--ID: 1734520487336-->
+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).
+<!--ID: 1734520487339-->
+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).
+<!--ID: 1734520487342-->
+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).
+<!--ID: 1734520487345-->
+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).
+<!--ID: 1734520487349-->
+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).
+<!--ID: 1734520487352-->
+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).
+<!--ID: 1734520487356-->
+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).
+<!--ID: 1734520487359-->
+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
 <!--ID: 1734374219332-->
 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).
+<!--ID: 1734520487363-->
+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).
+<!--ID: 1734520487368-->
+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).
+<!--ID: 1734520487372-->
+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).
+<!--ID: 1734520487376-->
+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).
+<!--ID: 1734520487381-->
+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).
+<!--ID: 1734520487384-->
+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).
+<!--ID: 1734520487388-->
+END%%
+
 ## Bibliography

 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).