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Area and induction.

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Joshua Potter 2024-11-09 19:23:36 -07:00
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37
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---
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).

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- [ ] Korean (Read 1 Story)

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---
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notes/algebra/sequences/index.md
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@ -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-->

2
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@ -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%%

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@ -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/).

54
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@ -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) {
return E;
}
int* func(int *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) {
return E[0];
}
int func(int *E) { 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) {
return E[i];
}
int func(int *E, int 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) {
return &E[2];
}
int* func(int *E) { return &E[2]; }
```
Back:
```asm

33
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@ -1360,24 +1360,40 @@ END%%
%%ANKI
Basic
What type is given to enumeration constants?
Back: `signed int`
Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
<!--ID: 1727022810819-->
What name is given to members of an enumeration?
Back: An enumeration constant.
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: 1730902791672-->
END%%
%%ANKI
Basic
What kind of integer expressions are `enum` constants limited to?
Back: Integer constant expressions.
What type is given to enumerated-type variables?
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?
Back: The ICE evaluates to an integer that fits within a `signed int`.
Besides being an ICE, what other condition is expected on enumeration constants?
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.

2
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@ -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%%

2
notes/combinatorics/permutations.md
View File

@ -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%%

2
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@ -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%%

113
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@ -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).

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

4
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@ -488,8 +488,8 @@ END%%
%%ANKI
Basic
What is wrong with asking "what is the scope of $x$ in $\lambda$-term $P$"?
Back: We should be asking about a $\lambda x$.
What is wrong with asking "what is the scope of $x$ in $\lambda x. P$"?
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%%

92
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@ -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.
* **Inductive case**: Prove $P(k) \Rightarrow P(k + 1)$ for all $k \geq n_0$.
Let $P(n)$ be a predicate depending on a number $n \in \mathbb{N}$. Assume that
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.
* **Inductive case**: Assume $P(k)$ is true for all $n_0 \leq k < n$. Then prove $P(n)$ is true.
* **Base case**: $P(n_0)$ is true for some $n_0 \geq 0$, and
* **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?
Back: It can be used to solve at least the same set of problems weak induction can.
Why makes strong induction "stronger" than weak induction?
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).

186
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@ -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).

38
notes/x86-64/directives.md Normal file
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@ -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.

16
notes/x86-64/instructions/index.md
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@ -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: