Pigeonhole principle, pointers.

2024-11-25 09:28:11 -07:00 · 2024-11-25 09:28:11 -07:00 · 677877e4ce
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    "Basic": [
--- a/notes/_journal/2024-11-25.md
+++ b/notes/_journal/2024-11-25.md
@ -0,0 +1,12 @@
+---
+title: "2024-11-25"
+---
+
+- [ ] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on the [[cardinality#Pigeonhole Principle|pigeonhole principle]].
+* Notes on pointers and function pointers.
--- a/notes/_journal/2024-11/2024-11-23.md
+++ b/notes/_journal/2024-11/2024-11-23.md
--- a/notes/_journal/2024-11/2024-11-24.md
+++ b/notes/_journal/2024-11/2024-11-24.md
@ -0,0 +1,11 @@
+---
+title: "2024-11-24"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on the [[derived#NULL|NULL]] macro.
--- a/notes/c17/types/derived.md
+++ b/notes/c17/types/derived.md
@ -528,6 +528,47 @@ Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co
 <!--ID: 1727957576041-->
 END%%

+Evaluation of an array `A` returns `&A[0]`, i.e. a [[#Pointers|pointer]] to the first array element. This is called **array decay**.
+
+%%ANKI
+Basic
+What is the effect of array decay?
+Back: Evaluation of an array `A` returns `&A[0]`.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953228-->
+END%%
+
+%%ANKI
+Basic
+What name is given to the implicit conversion of an array to a pointer?
+Back: Array decay.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953231-->
+END%%
+
+%%ANKI
+Basic
+According to Gustedt, what C feature explains why are there no "array values"?
+Back: Array-to-pointer decay.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953234-->
+END%%
+
+%%ANKI
+Basic
+Why can't arrays directly be made arguments to functions?
+Back: Because array arguments decay to pointers.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953237-->
+END%%
+
+%%ANKI
+Cloze
+In a function declaration, any array parameter rewrites to {a pointer}.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953240-->
+END%%
+
 #### Fixed-Length

 A fixed-length array (FLA) has a predetermined size. Their stack allocations can be computed at compilation time.
@ -1270,9 +1311,220 @@ Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co
 <!--ID: 1732397726962-->
 END%%

+### NULL
+
+The `NULL` macro refers to a **null pointer constant**, an ICE with value `0` or such an expression cast to type `void*`. The following table lists some valid values `NULL` can take on:
+
+| Expansion                  | Type                 |
+| -------------------------- | -------------------- |
+| `0U`                       | `unsigned`           |
+| `0`                        | `signed`             |
+| `\0`                       | `signed`             |
+| Enum constant of value `0` | `signed`             |
+| `0UL`                      | `unsigned long`      |
+| `0L`                       | `signed long`        |
+| `0ULL`                     | `unsigned long long` |
+| `0LL`                      | `signed long long`   |
+| `(void*)0`                 | `void*`              |
+
+%%ANKI
+Basic
+How are null pointer constants defined in terms of ICEs?
+Back: As any ICE with value `0` or such an expression cast to type `void*`.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732456644395-->
+END%%
+
+%%ANKI
+Basic
+What *must* the `NULL` macro expand to?
+Back: Any null pointer constant.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732456644434-->
+END%%
+
+%%ANKI
+Basic
+Which of the following members of the list are ICEs?
+```c
+0U, '\0', 0UL, (void*)0, 5LL
+```
+Back: `0U`, `\0`, and `0UL`.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732456644440-->
+END%%
+
+%%ANKI
+Basic
+Which of the following members of the list are null pointer constants?
+```c
+0U, '\0', 0UL, (void*)0, 5LL
+```
+Back: `0U`, `\0`, `0UL`, and `(void*)0`.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732456644446-->
+END%%
+
+%%ANKI
+Basic
+Which of the following members of the list could `NULL` be identical to?
+```c
+0U, '\0', 0UL, (void*)0, 5LL
+```
+Back: `0U`, `\0`, `0UL`, and `(void*)0`.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732456644454-->
+END%%
+
+%%ANKI
+Basic
+Which of the following members of the list are pointer constants?
+```c
+0U, '\0', 0UL, (void*)0, 5LL
+```
+Back: Just `(void*)0`.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732456644461-->
+END%%
+
+%%ANKI
+Basic
+Why does Gustedt discourage use of `NULL`?
+Back: The type of value it expands to is implementation-specific.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732456644469-->
+END%%
+
+%%ANKI
+Basic
+What is wrong with the following invocation?
+```c
+printf("%d, %p", 1, NULL);
+```
+Back: `NULL` may not refer to a pointer type.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732456644475-->
+END%%
+
+%%ANKI
+Basic
+What value must `NULL` have for the following to be correct?
+```c
+printf("%d, %p", 1, NULL);
+```
+Back: `(void*)0`
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732456644482-->
+END%%
+
+## Functions
+
+A function `f` without a following opening `(` is converted to a pointer to its start. This is called **function decay**.
+
+%%ANKI
+Basic
+What is the effect of function decay?
+Back: Evaluation of a function `f` without a following opening `(` is converted to a pointer to its start.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953243-->
+END%%
+
+%%ANKI
+Basic
+What name is given to the implicit conversion of a function to a pointer?
+Back: Function decay.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953247-->
+END%%
+
+%%ANKI
+Basic
+According to Gustedt, what C feature explains why are there no "function values"?
+Back: Function-to-pointer decay.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953250-->
+END%%
+
+%%ANKI
+Basic
+Why can't functions directly be made arguments to functions?
+Back: Because function arguments decay to pointers.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953255-->
+END%%
+
+%%ANKI
+Cloze
+{1:Function pointers} are to {2:`(...)`} whereas {2:pointers} are to {1:`[...]`}.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953260-->
+END%%
+
+%%ANKI
+Basic
+In what order are decays, dereferences, address ofs, and calls performed in the following?
+```c
+f(3);
+```
+Back: Decay, call.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953264-->
+END%%
+
+%%ANKI
+Basic
+In what order are decays, dereferences, address ofs, and calls performed in the following?
+```c
+(&f)(3);
+```
+Back: Address of, call.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953269-->
+END%%
+
+%%ANKI
+Basic
+In what order are decays, dereferences, address ofs, and calls performed in the following?
+```c
+(*f)(3);
+```
+Back: Decay, dereference, decay, call.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953273-->
+END%%
+
+%%ANKI
+Basic
+In what order are decays, dereferences, address ofs, and calls performed in the following?
+```c
+(*&f)(3);
+```
+Back: Address of, dereference, decay, call.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953277-->
+END%%
+
+%%ANKI
+Basic
+In what order are decays, dereferences, address ofs, and calls performed in the following?
+```c
+(&*f)(3);
+```
+Back: Decay, dereference, address of, call.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953281-->
+END%%
+
+%%ANKI
+Cloze
+{1:Pointers} refer to {2:arrays} whereas {2:function pointers} refer to {1:functions}.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1732551953285-->
+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 - 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).
-* Van der Linden, Peter. _Expert C Programming: Deep C Secrets_. Programming Languages / C. Mountain View, Cal.: SunSoft Pr, 1994.
+* Van der Linden, Peter. _Expert C Programming: Deep C Secrets_. Programming Languages / C. Mountain View, Cal.: SunSoft Pr, 1994.
--- a/notes/set/cardinality.md
+++ b/notes/set/cardinality.md
@ -6,7 +6,7 @@ tags:
  - set
 ---

-## Overview
+## Equinumerosity

 We say set $A$ is **equinumerous** to set $B$, written ($A \approx B$) if and only if there exists a [[functions#Injections|one-to-one]] function from $A$ [[functions#Surjections|onto]] $B$.

@ -58,7 +58,51 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1732295060366-->
 END%%

-## Equivalence Concept
+### Power Sets
+
+No set is equinumerous to its [[set/index#Power Set Axiom|power set]]. This is typically shown using a diagonalization argument.
+
+%%ANKI
+Basic
+What basic set operation produces a new set the original isn't equinumerous to?
+Back: The power set operation.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732541309202-->
+END%%
+
+%%ANKI
+Basic
+What kind of argument is typically made to prove no set is equinumerous to its power set?
+Back: A diagonalization argument.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732541309208-->
+END%%
+
+%%ANKI
+Basic
+Who is attributed the discovery of the diagonalization argument?
+Back: Georg Cantor.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732541309212-->
+END%%
+
+%%ANKI
+Basic
+Let $g \colon A \rightarrow \mathscr{P}A$. Using a diagonalization argument, what set is *not* in $\mathop{\text{ran}}(g)$?
+Back: $\{ x \in A \mid x \not\in g(x) \}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732541309216-->
+END%%
+
+%%ANKI
+Basic
+Let $g \colon A \rightarrow \mathscr{P}A$. *Why* isn't $B = \{x \in A \mid x \not\in g(x) \}$ in $\mathop{\text{ran}}(g)$?
+Back: For all $x \in A$, $x \in B \Leftrightarrow x \not\in g(x)$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732541309221-->
+END%%
+
+### Equivalence Concept

 For any sets $A$, $B$, and $C$:

@ -143,6 +187,150 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1732295060403-->
 END%%

+## Finiteness
+
+A set is **finite** if and only if it is equinumerous to some [[natural-numbers|natural number]]. Otherwise it is **infinite**.
+
+%%ANKI
+Basic
+How does Enderton define a finite set?
+Back: As a set equinumerous to some natural number.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231320-->
+END%%
+
+%%ANKI
+Basic
+How does Enderton define an infinite set?
+Back: As a set not equinumerous to any natural number.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231330-->
+END%%
+
+%%ANKI
+Basic
+Is $n \in \omega$ a finite set?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231336-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't $n \in \omega$ a finite set?
+Back: N/A. It is.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231342-->
+END%%
+
+%%ANKI
+Basic
+Is $\omega$ a finite set?
+Back: No.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231347-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't $\omega$ a finite set?
+Back: There is no natural number equinumerous to $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231353-->
+END%%
+
+### Pigeonhole Principle
+
+No natural number is equinumerous to a proper subset of itself. More generally, no finite set is equinumerous to a proper subset of itself.
+
+%%ANKI
+Basic
+How does Enderton state the pigeonhole principle for $\omega$?
+Back: No natural number is equinumerous to a proper subset of itself.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231358-->
+END%%
+
+%%ANKI
+Basic
+How does Enderton state the pigeonhole principle for finite sets?
+Back: No finite set is equinumerous to a proper subset of itself.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231364-->
+END%%
+
+%%ANKI
+Basic
+Let $m \in n \in \omega$. What principle precludes $m \approx n$?
+Back: The pigeonhole principle.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231369-->
+END%%
+
+%%ANKI
+Basic
+Let $S$ be a set and $n \in \omega$ such that $S \approx n$. For $m \in \omega$, when might $S \approx m$?
+Back: *Only* if $m = n$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231374-->
+END%%
+
+%%ANKI
+Basic
+What is the generalization of the pigeonhole principle for $\omega$?
+Back: The pigeonhole principle for finite sets.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231379-->
+END%%
+
+%%ANKI
+Basic
+What is the specialization of the pigeonhole principle for finite sets?
+Back: The pigeonhole principle for $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231385-->
+END%%
+
+%%ANKI
+Basic
+What name is given to the following theorem? $$\text{No finite set is equinumerous to a proper subset of itself.}$$
+Back: The pigeonhole principle.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231391-->
+END%%
+
+%%ANKI
+Basic
+Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. Is $f$ a bijection?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231396-->
+END%%
+
+%%ANKI
+Basic
+Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. *Why* must $f$ be surjective?
+Back: Otherwise $f$ is a bijection between $S$ and a proper subset of $S$, a contradiction.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231401-->
+END%%
+
+%%ANKI
+Basic
+Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. Is $f$ a bijection?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231407-->
+END%%
+
+%%ANKI
+Basic
+Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. *Why* must $f$ be injective?
+Back: Otherwise $f$ is a bijection between a proper subset of $S$ and $S$, a contradiction.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1732545231412-->
+END%%
+
 ## Bibliography

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