Area and induction.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

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     "algorithms/running-time.md": "5efc0791097d2c996f931c9046c95f65",
-    "algorithms/order-growth.md": "b2868f2523bb85f9e9c3fbfeb3753f0c",
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-    "combinatorics/permutations.md": "794822b96aeeadcb31e763f52ffb7b6e",
+    "combinatorics/permutations.md": "8eeafb451a6c2a46269b19790e163ae2",
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     "proofs/induction.md": "36ab5a92ae3cf9bb2def333dc41d79ff",
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     "x86-64/instructions/access.md": "4589d79f9915ffeeab7b75d2208b7917",
-    "x86-64/instructions/index.md": "df78f6d1d56da054606d374ca3282bf8",
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-    "calculus/bounds.md": "cbae7421eaa096cd17a2f9de079f593d",
+    "calculus/bounds.md": "f89de98e888b36af3b921fc0bdb1d281",
     "calculus/index.md": "5ee4d950533ae330ca5ef9e113fe87f3",
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-    "set/natural-numbers.md": "0fb40d1d2f7a0f9976afbacdff1c734b",
+    "set/natural-numbers.md": "6219f07da2d79fba29bb16bcd2f48ce2",
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     "c17/basic-types.md": "7c6653bf6dc24c2f2aa72fc95c4f7875",
-    "c17/types/simple.md": "b53fd365671848f09e6a0cd46e17c1bf",
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     "c17/types/enumerated.md": "e1f70a30677c776b7b44ac3e0ff4e76d",
-    "c17/types/derived.md": "89b86ecc2440bcb1aff13e1504efd1ba",
+    "c17/types/derived.md": "ac1651775cc1a75247da176627fcf421",
     "c17/types/basic.md": "5064e21e683c0218890058882e06b6f3",
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     "algebra/abs-val.md": "a47bc08db62304eb526d15ede3e300cf",
-    "data-structures/graphs.md": "93233f0fa6980b8b311de1660de3f3e2",
+    "data-structures/graphs.md": "0f47148557943dbca068c166d3e4169a",
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   "fields_dict": {
     "Basic": [

notes/_journal/2024-11-09.md

@@ -0,0 +1,13 @@
+---
+title: "2024-11-09"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Additional notes on alignment.
+* Continued reading "Calculus, Vol. 1" by Apostol.
+* Additional nodes on induction principles (weak, strong, well-ordering).

notes/_journal/2024-11/2024-11-05.md

@@ -2,8 +2,8 @@
 title: "2024-11-05"
 ---
 
-- [ ] Anki Flashcards
+- [x] Anki Flashcards
-- [ ] KoL
+- [x] KoL
 - [ ] OGS
 - [ ] Sheet Music (10 min.)
 - [ ] Korean (Read 1 Story)

notes/_journal/2024-11/2024-11-06.md

@@ -0,0 +1,9 @@
+---
+title: "2024-11-06"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/_journal/2024-11/2024-11-07.md

@@ -0,0 +1,9 @@
+---
+title: "2024-11-07"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/_journal/2024-11/2024-11-08.md

@@ -0,0 +1,9 @@
+---
+title: "2024-11-08"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/algebra/sequences/index.md

@@ -384,7 +384,7 @@ END%%
 %%ANKI
 Basic
 Why isn't $a_n = 2a_{n-1} + 3a_{n-2}$ a linear homogeneous recurrence relation with constant coefficients?
-Back: It is.
+Back: N/A. It is.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 Tags: algebra::polynomial
 <!--ID: 1714487050023-->

notes/algorithms/order-growth.md

@@ -1211,7 +1211,7 @@ END%%
 %%ANKI
 Basic
 What is the symmetric property of $\omega$-notation?
-Back: N/A
+Back: N/A.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1709752223553-->
 END%%

notes/c17/alignment.md

@@ -1,15 +1,29 @@
 ---
 title: Alignment
 TARGET DECK: Obsidian::STEM
-FILE TAGS: c17 x86-64
+FILE TAGS: c17::alignment x86-64
 tags:
   - c17
+  - x86-64
 ---
 
 ## Overview
 
 For a large class of modern ISAs, storage for basic C datatypes respect **self-alignmnet**. This means `char`s can start on any byte address, `short`s on any even address, 4-byte `int`s and `float`s must start on an address divisible by 4, and `double`s must start on an address divisible by 8. Likewise pointers are also self-aligned.
 
+Wasted space introduced solely for alignment purposes is referred to as **slop**.
+
+## Structures
+
+A `struct`'s **stride address** refers to the first address following the `struct` data that has the same alignment as the `struct`. In general the compiler adds various constraints to how a `struct` is layed out:
+
+1. The `struct`'s alignment follows that of its widest scalar member.
+	* This guarantees each member satisfies its own self-alignment requirement.
+2. The `struct` introduces trailing padding up to its stride address.
+	1. This ensures each element in an array satsifies its self-alignment requirement.
+
+The `sizeof` operator on a `struct` returns the total space used between its leading address and its stride address.
+
 %%ANKI
 Basic
 What does self-alignment refer to?
@@ -44,12 +58,20 @@ END%%
 
 %%ANKI
 Basic
-What addresses can a `double` be stored at?
+What addresses can a `double` be stored at (outside of a `struct`)?
 Back: Any address divisible by 8.
 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/).
 <!--ID: 1730831516938-->
 END%%
 
+%%ANKI
+Basic
+What addresses can a `double` be stored at (inside of a `struct`)?
+Back: Depending on compiler, addresses divisible by 4 or 8.
+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/).
+<!--ID: 1730988416659-->
+END%%
+
 %%ANKI
 Basic
 What addresses can a pointer be stored at?
@@ -97,6 +119,356 @@ Reference: Raymond, Eric. “The Lost Art of Structure Packing.” Accessed Nove
 <!--ID: 1730831516948-->
 END%%
 
+%%ANKI
+Basic
+Assume allocated order matches source order. How are the following variables aligned?
+```c
+char *p;
+char c;
+short x;
+```
+Back:
+```c
+char *p;     // 8 bytes
+char c;      // 1 byte
+char pad[1]; // 1 byte
+short x;     // 2 bytes
+```
+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/).
+<!--ID: 1730902219762-->
+END%%
+
+%%ANKI
+Basic
+Assume allocated order matches source order. How are the following variables aligned?
+```c
+long x;
+short c;
+float f;
+```
+Back:
+```c
+long x;      // 8 bytes
+short c;     // 2 bytes
+char pad[2]; // 2 bytes
+float f;     // 4 bytes
+```
+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/).
+<!--ID: 1730902219769-->
+END%%
+
+%%ANKI
+Basic
+Assume allocated order matches source order. What is the value of `N`?
+```c
+char c;
+char pad1[M];
+char *p;
+char pad2[N];
+int x;
+```
+Back: $0$
+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/).
+<!--ID: 1730902219773-->
+END%%
+
+%%ANKI
+Basic
+Assume allocated order matches source order. What is the value of `M`?
+```c
+char c;
+char pad1[M];
+char *p;
+char pad2[N];
+int x;
+```
+Back: Between $0$ and $7$ inclusive.
+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/).
+<!--ID: 1730902219776-->
+END%%
+
+%%ANKI
+Basic
+Consider `short A[N]`. What is the internal padding of this array?
+Back: $0$, i.e. `A` has no internal padding.
+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/).
+<!--ID: 1730902219780-->
+END%%
+
+%%ANKI
+Basic
+Consider `char* A[N]`. What is the internal padding of this array?
+Back: $0$, i.e. `A` has no internal padding.
+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/).
+<!--ID: 1730902219784-->
+END%%
+
+%%ANKI
+Basic
+What is the alignment of a `struct` instance?
+Back: That of its widest scalar member.
+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/).
+<!--ID: 1730902219788-->
+END%%
+
+%%ANKI
+Basic
+*Why* is the alignment of a `struct` instance that of its widest scalar member?
+Back: It is an easy way to ensure all members are self-aligned.
+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/).
+<!--ID: 1730902219792-->
+END%%
+
+%%ANKI
+Basic
+*Why* does a `struct` sometimes introduce trailing padding?
+Back: To ensure each element of an array of these `struct`s is self-aligned.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1731165663191-->
+END%%
+
+%%ANKI
+Cloze
+The address of a `struct` is the same as its {first member}.
+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/).
+<!--ID: 1730902219797-->
+END%%
+
+%%ANKI
+Basic
+How much leading padding does a `struct` instance have?
+Back: $0$
+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/).
+<!--ID: 1730902219802-->
+END%%
+
+%%ANKI
+Basic
+How are members of the following `struct` aligned?
+```c
+struct foo {
+  char *p;
+  char c;
+  long x;
+};
+```
+Back:
+```c
+struct foo {
+  char *p;     // 8 bytes
+  char c;      // 1 byte
+  char pad[7]; // 7 bytes
+  long x;      // 8 bytes
+};
+```
+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/).
+<!--ID: 1730902219809-->
+END%%
+
+%%ANKI
+Basic
+How are members of the following `struct` aligned?
+```c
+struct foo {
+  char c;
+  char *p;
+  long x;
+};
+```
+Back:
+```c
+struct foo {
+  char c;      // 1 byte
+  char pad[7]; // 7 bytes
+  char *p;     // 8 bytes
+  long x;      // 8 bytes
+};
+```
+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/).
+<!--ID: 1730902219814-->
+END%%
+
+%%ANKI
+Cloze
+A `struct`'s {stride address} is {the first address following the `struct`'s data with the same alignment as the `struct`}.
+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/).
+<!--ID: 1730902219820-->
+END%%
+
+%%ANKI
+Basic
+What is the result of `sizeof(struct foo)`?
+```c
+struct foo {
+  char *p;
+  char c;
+};
+```
+Back: $16$
+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/).
+<!--ID: 1730902219825-->
+END%%
+
+%%ANKI
+Basic
+What is the result of `sizeof(struct foo)`?
+```c
+struct foo {
+  short s;
+  char c;
+};
+```
+Back: $4$
+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/).
+<!--ID: 1730902219830-->
+END%%
+
+%%ANKI
+Basic
+What "hidden" space is returned by `sizeof` on `struct` types?
+Back: Slop.
+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/).
+<!--ID: 1730902219834-->
+END%%
+
+%%ANKI
+Basic
+How are members of the following `struct` aligned?
+```c
+struct foo {
+  char c;
+  struct bar {
+    char *p;
+    short x;
+  } bar;
+};
+```
+Back:
+```c
+struct foo {
+  char c;         // 1 byte
+  char pad[7];    // 7 bytes
+  struct bar {
+    char *p;      // 8 bytes
+    short x;      // 2 bytes
+    char pad2[6]; // 6 bytes
+  } bar;
+};
+```
+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/).
+<!--ID: 1730902219840-->
+END%%
+
+%%ANKI
+Basic
+How are members of the following `struct` aligned?
+```c
+struct foo {
+  char c;
+  struct bar {
+    short x;
+  } bar;
+};
+```
+Back:
+```c
+struct foo {
+  char c;         // 1 byte
+  char pad[1];    // 1 byte
+  struct bar {
+    short x;      // 2 bytes
+  } bar;
+};
+```
+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/).
+<!--ID: 1730902219846-->
+END%%
+
+%%ANKI
+Basic
+What is the alignment of a nested `struct`?
+Back: That of its widest scalar member.
+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/).
+<!--ID: 1730902219851-->
+END%%
+
+%%ANKI
+Basic
+The `sizeof` operator on `struct`s returns the space used between what two addresses?
+Back: The `struct`'s leading address and stride address.
+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/).
+<!--ID: 1731165663196-->
+END%%
+
+%%ANKI
+Basic
+What is the offset of field `u` in the following?
+```c
+struct foo {
+  char u;
+  int v;
+};
+```
+Back: `0`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1731165663205-->
+END%%
+
+%%ANKI
+Basic
+What is the offset of field `v` in the following?
+```c
+struct foo {
+  char u;
+  int v;
+};
+```
+Back: `4`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1731165663209-->
+END%%
+
+## Unions
+
+The `sizeof` operator on a `union` returns that of its widest member.
+
+%%ANKI
+Basic
+The `sizeof` operator on `union`s returns what?
+Back: The `sizeof` that of its widest member.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1731165663214-->
+END%%
+
+%%ANKI
+Basic
+What is the offset of field `u` in the following?
+```c
+union foo {
+  char u;
+  int v;
+};
+```
+Back: `0`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1731165663237-->
+END%%
+
+%%ANKI
+Basic
+What is the offset of field `v` in the following?
+```c
+union foo {
+  char u;
+  int v;
+};
+```
+Back: `0`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1731165663242-->
+END%%
+
 ## Bibliography
 
+* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 * 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/).

notes/c17/types/derived.md

@@ -825,7 +825,7 @@ END%%
 
 %%ANKI
 Basic
-Suppose a pointer logically evaluates to `true`. Why can't we use it?
+Suppose a pointer logically evaluates to `true`. Why might it still be unsafe to use?
 Back: Logical evaluation can't distinguish valid pointers from indeterminate pointers.
 Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
 <!--ID: 1730755551598-->
@@ -851,7 +851,7 @@ END%%
 Basic
 How is `r->field` equivalently written using `*`?
 Back: `(*r).field`
-Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1730757470069-->
 END%%
 
@@ -859,7 +859,7 @@ END%%
 Basic
 How is `(*r).field` more compactly written?
 Back: `r->field`
-Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1730757492874-->
 END%%
 
@@ -912,56 +912,56 @@ END%%
 
 %%ANKI
 Basic
-Given the following, how can we equivalent write `A[1][2]` using pointer arithmetic?
+How do we rewrite the return statement using pointer arithmetic?
 ```c
-int main() {
+int func() {
   int A[3][4];
-  return 0;
+  return A[1][2];
 }
 ```
-Back: `A + 6`
+Back: `return *(A + 6);`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1730745799783-->
 END%%
 
 %%ANKI
 Basic
-Given the following, how can we equivalent write `A[2][0]` using pointer arithmetic?
+How do we rewrite the return statement using pointer arithmetic?
 ```c
-int main() {
+int func() {
   int A[3][4];
-  return 0;
+  return A[2][0];
 }
 ```
-Back: `A + 8`
+Back: `return *(A + 8);`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1730745799787-->
 END%%
 
 %%ANKI
 Basic
-Given the following, how can we equivalent write `A[1][2]` using pointer arithmetic?
+How do we rewrite the return statement using pointer arithmetic?
 ```c
-int main() {
+int func() {
   int A[4][3];
-  return 0;
+  return A[1][2];
 }
 ```
-Back: `A + 5`
+Back: `return *(A + 5);`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1730745799789-->
 END%%
 
 %%ANKI
 Basic
-Given the following, how can we equivalent write `A[1][0]` using pointer arithmetic?
+How do we rewrite the return statement using pointer arithmetic?
 ```c
-int main() {
+int func() {
   int A[4][3];
-  return 0;
+  return A[1][0];
 }
 ```
-Back: `A + 3`
+Back: `return *(A + 3);`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1730745799791-->
 END%%
@@ -970,9 +970,7 @@ END%%
 Basic
 How is the `return` statement likely translated to x86-64?
 ```c
-int* example(int *E) {
+int* func(int *E) { return E; }
-  return E;
-}
 ```
 Back:
 ```asm
@@ -987,9 +985,7 @@ END%%
 Basic
 How is the `return` statement likely translated to x86-64?
 ```c
-int example(int *E) {
+int func(int *E) { return E[0]; }
-  return E[0];
-}
 ```
 Back:
 ```asm
@@ -1004,9 +1000,7 @@ END%%
 Basic
 How is the `return` statement likely translated to x86-64?
 ```c
-int example(int *E, int i) {
+int func(int *E, int i) { return E[i]; }
-  return E[i];
-}
 ```
 Back:
 ```asm
@@ -1021,9 +1015,7 @@ END%%
 Basic
 How is the `return` statement likely translated to x86-64?
 ```c
-int example(int *E) {
+int* func(int *E) { return &E[2]; }
-  return &E[2];
-}
 ```
 Back:
 ```asm

notes/c17/types/simple.md

@@ -1360,24 +1360,40 @@ END%%
 
 %%ANKI
 Basic
-What type is given to enumeration constants?
+What name is given to members of an enumeration?
-Back: `signed int`
+Back: An enumeration constant.
-Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+Reference: “ISO: Programming Languages - C,” April 12, 2011, [https://port70.net/~nsz/c/c11/n1570.pdf](https://port70.net/~nsz/c/c11/n1570.pdf).
-<!--ID: 1727022810819-->
+<!--ID: 1730902791672-->
 END%%
 
 %%ANKI
 Basic
-What kind of integer expressions are `enum` constants limited to?
+What type is given to enumerated-type variables?
-Back: Integer constant expressions.
+Back: N/A. This is compiler dependent.
+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/).
+<!--ID: 1730902856916-->
+END%%
+
+%%ANKI
+Basic
+What type is *usually* given to enumerated-type variables?
+Back: `int`
+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/).
+<!--ID: 1730902856922-->
+END%%
+
+%%ANKI
+Basic
+What kind of integer expressions are enumeration constants limited to?
+Back: Integer constant expressions with value representable as an `int`.
 Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
 <!--ID: 1727022810827-->
 END%%
 
 %%ANKI
 Basic
-Besides being an ICE, what other condition is expected on `enum` values?
+Besides being an ICE, what other condition is expected on enumeration constants?
-Back: The ICE evaluates to an integer that fits within a `signed int`.
+Back: The ICE evaluates to an integer that fits within an `int`.
 Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
 <!--ID: 1727022810833-->
 END%%
@@ -1402,4 +1418,5 @@ END%%
 
 * “ISO: Programming Languages - C,” April 12, 2011, [https://port70.net/~nsz/c/c11/n1570.pdf](https://port70.net/~nsz/c/c11/n1570.pdf).
 * Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+* 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/).
 * Van der Linden, Peter. _Expert C Programming: Deep C Secrets_. Programming Languages / C. Mountain View, Cal.: SunSoft Pr, 1994.

notes/calculus/bounds.md

@@ -29,7 +29,7 @@ END%%
 %%ANKI
 Basic
 Suppose $S \subseteq \mathbb{R}$ is bounded below by $B$. What property does set $S$ exhibit?
-Back: $S$ is nonempty..
+Back: $S$ is nonempty.
 Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
 <!--ID: 1724115335387-->
 END%%

notes/combinatorics/permutations.md

@@ -347,7 +347,7 @@ Basic
 ```
 [ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]
 ```
-Back: N/A
+Back: N/A.
 Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
 <!--ID: 1709756610403-->
 END%%

notes/data-structures/graphs.md

@@ -30,7 +30,7 @@ END%%
 %%ANKI
 Basic
 What are the two standard ways of representing graphs in memory?
-Back: The adjacency-list and adjacency-matrix representation.
+Back: The adjacency-list and adjacency-matrix representations.
 Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1724614579422-->
 END%%

notes/geometry/cartesian.md

@@ -0,0 +1,113 @@
+---
+title: Cartesian Coordinate System
+TARGET DECK: Obsidian::STEM
+FILE TAGS: geometry::coordinates
+tags:
+  - geometry
+---
+
+## Overview
+
+In plane analytic geometry, the **Cartesian coordinate system** uniquely specifies a point by a pair of real numbers called its **coordinates**. These coordinates represent signed distances to the point from two fixed perpendicular oriented lines called the **axes**. The point where the axes meet is called the **origin** and have coordinates $\langle 0, 0 \rangle$.
+
+%%ANKI
+Cloze
+The {$x$-coordinate} of a point is sometimes called its {abscissa).
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731184865785-->
+END%%
+
+%%ANKI
+Cloze
+The {$y$-coordinate} of a point is sometimes called its {ordinate).
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731184865791-->
+END%%
+
+%%ANKI
+Cloze
+The {origin} of a Cartesian coordinate system has coordinates $\langle 0, 0 \rangle$.
+Reference: “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).
+<!--ID: 1731184876657-->
+END%%
+
+%%ANKI
+Basic
+Consider point $\langle x, y \rangle$. When does this point lie in the first quadrant?
+Back: When $x > 0$ and $y > 0$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731184865794-->
+END%%
+
+%%ANKI
+Basic
+Consider point $\langle x, y \rangle$. When does this point lie in the second quadrant?
+Back: When $x < 0$ and $y > 0$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731184865796-->
+END%%
+
+%%ANKI
+Basic
+Consider point $\langle x, y \rangle$. When does this point lie in the fourth quadrant?
+Back: When $x > 0$ and $y < 0$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731184865799-->
+END%%
+
+%%ANKI
+Basic
+Consider point $\langle x, y \rangle$. When does this point lie in the third quadrant?
+Back: When $x < 0$ and $y < 0$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731184865801-->
+END%%
+
+%%ANKI
+Basic
+The "vertical line test" of a Cartesian coordinate system is used to determine what?
+Back: Whether the tested graph depicts a function or not.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731185808236-->
+END%%
+
+%%ANKI
+Basic
+In Cartesian coordinate systems, why does the vertical line test work?
+Back: A function is single-valued. A vertical line that intersects a graph multiple times immediately contradicts this.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731185808243-->
+END%%
+
+## Cartesian Equations
+
+An equation that completely characters a figure within the Cartesian coordinate system is called a **Cartesian equation**.
+
+%%ANKI
+Basic
+What is a Cartesian equation?
+Back: An equation that completely characterizes a figure within the Cartesian coordinate system.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731185808246-->
+END%%
+
+%%ANKI
+Basic
+What is the Cartesian equation of a circle centered around the origin with radius $r$?
+Back: $x^2 + y^2 = r^2$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731185808249-->
+END%%
+
+%%ANKI
+Basic
+What figure does the following Cartesian equation characterize? $x^2 + y^2 = r^2$
+Back: A circle with radius $r$ centered around the origin.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1731185808252-->
+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).
+* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).

notes/geometry/index.md

@@ -0,0 +1,3 @@
+---
+title: Geometry
+---

notes/lambda-calculus/index.md

@@ -488,8 +488,8 @@ END%%
 
 %%ANKI
 Basic
-What is wrong with asking "what is the scope of $x$ in $\lambda$-term $P$"?
+What is wrong with asking "what is the scope of $x$ in $\lambda x. P$"?
-Back: We should be asking about a $\lambda x$.
+Back: We should be asking what the scope of $\lambda x$ is.
 Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
 <!--ID: 1716745016007-->
 END%%

notes/proofs/index.md

@@ -295,12 +295,16 @@ END%%
 
 ## Induction
 
-Let $P(n)$ be a predicate. To prove $P(n)$ is true for all $n \geq n_0$, we prove:
+### Weak Induction
 
-* **Base case**: Prove $P(n_0)$ is true. This is usually done directly.
+Let $P(n)$ be a predicate depending on a number $n \in \mathbb{N}$. Assume that
-* **Inductive case**: Prove $P(k) \Rightarrow P(k + 1)$ for all $k \geq n_0$.
 
-Within the inductive case, $P(k)$ is known as the **inductive hypothesis**.
+* **Base case**: $P(n_0)$ is true for some $n_0 \geq 0$, and
+* **Inductive case**: for all $k \geq n_0$, $P(k) \Rightarrow P(k + 1)$.
+
+Then $P(n)$ is true for all $n \geq n_0$.
+
+Within the inductive case, $P(k)$ is known as the **inductive hypothesis**. The formal justification of proof by induction is intimately tied to the idea of [[natural-numbers#Inductive Sets|inductive sets]].
 
 %%ANKI
 Cloze
@@ -342,7 +346,7 @@ END%%
 
 %%ANKI
 Basic
-What proposition is typically proven in the base case of an inductive proof?
+What proposition is typically proven in the base case of a weak induction proof?
 Back: $P(n_0)$ for some $n_0 \geq 0$.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 <!--ID: 1714530152713-->
@@ -350,7 +354,7 @@ END%%
 
 %%ANKI
 Basic
-What proposition is typically proven in the inductive case of an inductive proof?
+What proposition is typically proven in the inductive case of a weak induction proof?
 Back: $P(k) \Rightarrow P(k + 1)$ for all $k \geq n_0$.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 <!--ID: 1714530152718-->
@@ -396,35 +400,31 @@ END%%
 
 ### Strong Induction
 
-Strong induction expands the induction hypothesis. Let $P(n)$ be a predicate. To prove $P(n)$ is true for all $n \geq n_0$, we prove:
+Let $P(n)$ be a predicate depending on a number $n \in \mathbb{N}$. Assume that
 
-* **Base case**: Prove $P(n_0)$ is true. This is usually done directly.
+* **Base case**: $P(n_0)$ is true for some $n_0 \geq 0$, and
-* **Inductive case**: Assume $P(k)$ is true for all $n_0 \leq k < n$. Then prove $P(n)$ is true.
+* **Inductive case**: for all $k \geq n_0$, $P(n_0) \land P(n_0 + 1) \land \cdots \land P(k) \Rightarrow P(k + 1)$.
+
+Then $P(n)$ is true for all $n \geq n_0$.
+
+The formal justification of proof by induction is intimately tied to the idea of [[natural-numbers#Inductive Sets|inductive sets]] and the [[natural-numbers#Well-Ordering Principle|well-ordering principle]].
 
 %%ANKI
 Basic
 Using typical identifiers, what is the inductive hypothesis of $P(n)$ using strong induction?
-Back: Assume $P(k)$ for all $k < n$.
+Back: Assume $P(k)$ for all $n_0 \leq k < n$.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 <!--ID: 1714574131949-->
 END%%
 
 %%ANKI
 Basic
-Why is strong induction considered stronger than weak induction?
+Why makes strong induction "stronger" than weak induction?
-Back: It can be used to solve at least the same set of problems weak induction can.
+Back: It gives more propositions in the antecedent of the inductive case.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 <!--ID: 1714574131955-->
 END%%
 
-%%ANKI
-Basic
-What negation is introduced to explain why the strong induction assumption is valid?
-Back: If $P(n)$ is not true for all $n$, there exists a *first* $n_0$ for which $\neg P(n_0)$.
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
-<!--ID: 1714574131963-->
-END%%
-
 %%ANKI
 Basic
 What distinguishes the base case of weak and strong induction proofs?
@@ -433,6 +433,58 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1714574131969-->
 END%%
 
+%%ANKI
+Basic
+How is the following strong induction clause rewritten to use weak induction? $$P(0) \land P(1) \land \cdots \land P(k) \Rightarrow P(k + 1)$$
+Back: As $Q(k) \Rightarrow Q(k + 1)$ where $Q(n) = P(0) \land P(1) \land \cdots \land P(n)$ for all $n \in \omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731203636959-->
+END%%
+
+%%ANKI
+Basic
+How is the following weak induction clause rewritten to use strong induction? $$P(k) \Rightarrow P(k + 1)$$
+Back: As $P(n_0) \land P(n_0 + 1) \land \cdots \land P(k) \Rightarrow P(k + 1)$ for some $0 \leq n_0$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731203636963-->
+END%%
+
+### Well-Ordering Principle
+
+This is covered [[natural-numbers#Well-Ordering Principle|here]]. It is equivalent to weak and strong induction.
+
+%%ANKI
+Basic
+What are the three most commonly used principles of induction?
+Back: Weak induction, strong induction, and well-ordering.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731203636955-->
+END%%
+
+%%ANKI
+Basic
+Why are names "weak" and "strong" induction a misnomer?
+Back: Weak and strong induction are logically equivalent.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731204485580-->
+END%%
+
+%%ANKI
+Basic
+What is PMI an acronym for?
+Back: The **p**rinciple of **m**athematical **i**nduction.
+Reference: N/A.
+<!--ID: 1731205303107-->
+END%%
+
+%%ANKI
+Basic
+What is WOP an acronym for?
+Back: The **w**ell-**o**rdering **p**rinciple.
+Reference: N/A.
+<!--ID: 1731205303114-->
+END%%
+
 ## Bibliography
 
 * “Constructive Proof,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Constructive_proof](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).

notes/set/natural-numbers.md

@@ -1149,6 +1149,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1729804914186-->
 END%%
 
+%%ANKI
+Basic
+What is the order-preserving property of addition on $\omega$ in FOL?
+Back: $\forall m, n, p \in \omega, m \in n \Leftrightarrow m + p \in n + p$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731170040087-->
+END%%
+
 ### Multiplication
 
 For each $m \in \omega$, there exists (by the recursion theorem) a unique function $M_m \colon \omega \rightarrow \omega$ such that for all $n \in \omega$, $$\begin{align*} M_m(0) & = 0, \\ M_m(n^+) & = M_m(n) + m \end{align*}$$
@@ -1227,6 +1235,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1729804914195-->
 END%%
 
+%%ANKI
+Basic
+What is the order-preserving property of multiplication on $\omega$ in FOL?
+Back: $\forall m, n, p \in \omega, m \in n \Leftrightarrow m \cdot p^+ \in n \cdot p^+$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731170040116-->
+END%%
+
 ### Exponentiation
 
 For each $m \in \omega$, there exists (by the recursion theorem) a unique function $E_m \colon \omega \rightarrow \omega$ such that for all $n \in \omega$, $$\begin{align*} E_m(0) & = 1, \\ E_m(n^+) & = E_m(n) \cdot m \end{align*}$$
@@ -1307,7 +1323,11 @@ END%%
 
 ## Ordering
 
-For natural numbers $m$ and $n$, define $m$ to be **less than $n$** if and only if $m \in n$. It follows $m \in n \Leftrightarrow m^+ \in n^+$.
+For natural numbers $m$ and $n$, define $m$ to be **less than $n$** if and only if $m \in n$. The following biconditionals hold true:
+
+* $m \in n \Leftrightarrow m^+ \in n^+$
+* $m \in n \Leftrightarrow m \subset n$
+* $m \underline{\in} n \Leftrightarrow m \subseteq n$
 
 %%ANKI
 Basic
@@ -1363,6 +1383,38 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1730118488846-->
 END%%
 
+%%ANKI
+Basic
+Let $m, n \in \omega$. What is the strict analog of $m \underline{\in} n \Leftrightarrow m \subseteq n$?
+Back: $m \in n \Leftrightarrow m \subset n$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731168085673-->
+END%%
+
+%%ANKI
+Basic
+Let $m, n \in \omega$. What is the non-strict analog of $m \in n \Leftrightarrow m \subset n$?
+Back: $m \underline{\in} n \Leftrightarrow m \subseteq n$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731168085679-->
+END%%
+
+%%ANKI
+Basic
+In what three ways does Enderton denote strict ordering of the natural numbers?
+Back: $\in$, $\subset$, and $<$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731170040122-->
+END%%
+
+%%ANKI
+Basic
+In what three ways does Enderton denote non-strict ordering of the natural numbers?
+Back: $\underline{\in}$, $\subseteq$, and $\leq$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731170040128-->
+END%%
+
 %%ANKI
 Basic
 What does the trichotomy law for $\omega$ state?
@@ -1371,6 +1423,138 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1730118488850-->
 END%%
 
+%%ANKI
+Basic
+Let $m, n \in \omega$. If $m \in n$, why is it that $m \subseteq n$?
+Back: Because $n$ is a transitive set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731168085682-->
+END%%
+
+%%ANKI
+Basic
+Let $m, n \in \omega$. If $m \in n$, why is it that $m \subset n$?
+Back: Because $n$ is a transitive set and no natural number is a member of itself.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731168085685-->
+END%%
+
+%%ANKI
+Basic
+What does Enterton describe is the typical way of using trichotomy in proofs?
+Back: Showing that two of the three possibilities is false.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731168085688-->
+END%%
+
+### Well-Ordering Principle
+
+Let $A$ be a nonempty subset of $\omega$. Then there is some $m \in A$ such that $m \underline{\in} n$ for all $n \in A$.
+
+%%ANKI
+Basic
+What does the well-ordering principle state?
+Back: Every nonempty subset of $\omega$ has a least element.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731200524848-->
+END%%
+
+%%ANKI
+Basic
+How is the well-ordering principle stated in FOL?
+Back: $\forall A \subseteq \omega, A \neq \varnothing \Rightarrow \exists m \in A, \forall n \in A, m \underline{\in} n$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731200524851-->
+END%%
+
+%%ANKI
+Basic
+Let $A$ be a set of $\omega$. What condition is necessary for $A$ to have a least element?
+Back: $A \neq \varnothing$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731200524854-->
+END%%
+
+%%ANKI
+Basic
+What principle states every nonempty subset of $\omega$ has a least element?
+Back: The well-ordering principle.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731200524857-->
+END%%
+
+%%ANKI
+Basic
+What principle states every nonempty subset of $\omega$ has a greatest element?
+Back: N/A. This is not true.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731200524861-->
+END%%
+
+%%ANKI
+Basic
+Suppose $A$ is a subset of $\omega$ without a least element. What can be said about $A$?
+Back: $A = \varnothing$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731200524864-->
+END%%
+
+%%ANKI
+Basic
+*Why* is there no function $f \colon \omega \rightarrow \omega$ such that $f(n^+) \in f(n)$ for all $n \in \omega$?
+Back: $\mathop{\text{ran}} f$ would violate the well-ordering principle.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731200524868-->
+END%%
+
+%%ANKI
+Basic
+The following is a FOL representation of what principle?
+$$\forall A \subseteq \omega, A \neq \varnothing \Rightarrow \exists m \in A, \forall n \in A, m \underline{\in} n$$
+Back: The well-ordering principle for $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731203636938-->
+END%%
+
+%%ANKI
+Basic
+How can we show set $S$ coincides with $\omega$ using the well-ordering principle?
+Back: By showing $\omega - S$ has no least element.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731204485586-->
+END%%
+
+### Strong Induction Principle
+
+Let $A$ be a subset of $\omega$ and assume that for every $n \in \omega$, $$\text{if every number less than } n \text{ is in } A, \text{then } n \in A.$$
+Then $A = \omega$.
+
+%%ANKI
+Basic
+Let $A \subseteq \omega$. The strong induction principle for $\omega$ assumes what about every $n \in \omega$?
+Back: If every number less than $n$ is in $A$, then $n \in A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731203636943-->
+END%%
+
+%%ANKI
+Basic
+The following is a FOL representation of what principle?
+$$[\forall A \subseteq \omega, 0 \in A \land (\forall n \in \omega, n^+ \in \omega)] \Rightarrow A = \omega$$
+Back: The weak induction principle for $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731203636947-->
+END%%
+
+%%ANKI
+Basic
+The following is a FOL representation of what principle?
+$$[\forall A \subseteq \omega, \forall n \in \omega, (\forall m \in n, m \in A) \Rightarrow n \in A] \Rightarrow A = \omega$$
+Back: The strong induction principle for $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1731203636951-->
+END%%
+
 ## Bibliography
 
 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

notes/x86-64/directives.md

@@ -0,0 +1,38 @@
+---
+title: Directives
+TARGET DECK: Obsidian::STEM
+FILE TAGS: x86-64
+tags:
+  - x86-64
+---
+
+## Directives
+
+Assembly lines with a leading `.` are called **directives**. They are used to indicate some additional information about the program.
+
+%%ANKI
+Basic
+What term describes assembly lines with a leading `.`?
+Back: Directives.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1710959313822-->
+END%%
+
+%%ANKI
+Basic
+Assembly directives are important for what two programs?
+Back: The assembler and the linker.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1710959313826-->
+END%%
+
+%%ANKI
+Cloze
+The {`.align N`} directive ensures {data following it starts with an address that is a multiple of `N`}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1731165663161-->
+END%%
+
+## Bibliography
+
+* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

notes/x86-64/instructions/index.md

@@ -44,22 +44,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1710959313818-->
 END%%
 
-%%ANKI
-Basic
-What term describes assembly lines with a leading `.`?
-Back: Directives.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1710959313822-->
-END%%
-
-%%ANKI
-Basic
-Assembly directives are important for what two programs?
-Back: The assembler and the linker.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1710959313826-->
-END%%
-
 ## Instruction Classes
 
 There are three types of operands: