From 3b87d57d8924def1a0050d6579de744a7957cfbe Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Sat, 23 Nov 2024 10:03:08 -0700 Subject: [PATCH] Axioms of area. --- .../plugins/obsidian-to-anki-plugin/data.json | 9 +- notes/_journal/2024-11-23.md | 9 + notes/_journal/{ => 2024-11}/2024-11-22.md | 0 notes/geometry/area.md | 341 ++++++++++++++++++ notes/geometry/cartesian.md | 24 ++ notes/geometry/index.md | 24 ++ 6 files changed, 404 insertions(+), 3 deletions(-) create mode 100644 notes/_journal/2024-11-23.md rename notes/_journal/{ => 2024-11}/2024-11-22.md (100%) create mode 100644 notes/geometry/area.md diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 4d24aee..d7c29c6 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -930,8 +930,8 @@ "_journal/2024-11/2024-11-08.md": "806bbade5f8339579287687f9433334e", "_journal/2024-11/2024-11-07.md": "434ec3f15d7065ea740127aa8477dd17", "x86-64/directives.md": "019c1c1d04efb26c3e8758aac4543cc7", - "geometry/cartesian.md": "b7003f70ab4822aa6eb4b84ba35f6e65", - "geometry/index.md": "679dcd097f4bebe417828c695444c88c", + "geometry/cartesian.md": "52eed93bf2d456ffab76e73d1031187c", + "geometry/index.md": "130185982889f115c9415f14c4424848", "_journal/2024-11-10.md": "5478337fd2017b99d0b359713a511e66", "_journal/2024-11/2024-11-09.md": "46f3a640223ef533f4523837b67b57c3", "_journal/2024-11-18.md": "5567592053951cee80450cf582df270a", @@ -950,7 +950,10 @@ "_journal/2024-11/2024-11-21.md": "951b6034d60a40dbd8201c50abf0dbb9", "_journal/2024-11/2024-11-20.md": "951b6034d60a40dbd8201c50abf0dbb9", "_journal/2024-11/2024-11-19.md": "d879f57154cb27cb168eb1f1f430e312", - "set/cardinality.md": "499e758bc0929af06736fa2974aade60" + "set/cardinality.md": "499e758bc0929af06736fa2974aade60", + "geometry/area.md": "3c6e53a64ad3150d8f81f6e4a63da61a", + "_journal/2024-11-23.md": "cd8db62b2aa0d67d50c0aa8257956732", + "_journal/2024-11/2024-11-22.md": "51117030e2364dbce3a8d507dead86ae" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-11-23.md b/notes/_journal/2024-11-23.md new file mode 100644 index 0000000..5ec64b9 --- /dev/null +++ b/notes/_journal/2024-11-23.md @@ -0,0 +1,9 @@ +--- +title: "2024-11-23" +--- + +- [ ] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) \ No newline at end of file diff --git a/notes/_journal/2024-11-22.md b/notes/_journal/2024-11/2024-11-22.md similarity index 100% rename from notes/_journal/2024-11-22.md rename to notes/_journal/2024-11/2024-11-22.md diff --git a/notes/geometry/area.md b/notes/geometry/area.md new file mode 100644 index 0000000..43cd764 --- /dev/null +++ b/notes/geometry/area.md @@ -0,0 +1,341 @@ +--- +title: Area +TARGET DECK: Obsidian::STEM +FILE TAGS: geometry::area +tags: + - calculus + - geometry +--- + +## Overview + +**Area** is a **set function** mapping from a class of so-called **measurable** sets $\mathscr{M}$ into the real numbers. + +%%ANKI +Basic +What is a set function? +Back: A function mapping a collection of sets to real numbers. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What is the first set function Apostol introduces? +Back: Area. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What kind of mathematical entity is area? +Back: A function. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What is the domain of the area function? +Back: The class of measurable sets. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What is the codomain of the area function? +Back: The real numbers. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What is the "function signature" of the area function $a$? +Back: $a \colon \mathscr{M} \rightarrow \mathbb{R}$ where $\mathscr{M}$ is the class of measurable sets. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What does Apostol mean by a measurable set? +Back: A set that can be ascribed an area. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +## Axioms + +We assume there exists a class $\mathscr{M}$ of measurable sets in the plane and a set function $a$, whose domain is $\mathscr{M}$, with the following six properties: + +### Nonnegative Property + +For each $S \in \mathscr{M}$, $a(S) \geq 0$. + +%%ANKI +Basic +What does the nonnegative property of area state? +Back: For every set $S \in \mathscr{M}$, $a(S) \geq 0$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +State the nonnegative property of area in FOL. +Back: $\forall S \in \mathscr{M}, a(S) \geq 0$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $a$ is an area function and $S \in \mathscr{M}$. Why can't $a(S) = -1$? +Back: This violates the nonnegative property of $a$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +### Additive Property + +If $S, T \in \mathscr{M}$, then $S \cup T$ and $S \cap T$ are in $\mathscr{M}$. Also $$a(S \cup T) = a(S) + a(T) - a(S \cap T).$$ + +Notice this last formulation is a special case of [[inclusion-exclusion|PIE]]. + +%%ANKI +Basic +Suppose $S, T \in \mathscr{M}$. What set(s) does the additive property of area state are also in $\mathscr{M}$? +Back: $S \cup T$ and $S \cap T$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $a$ is an area function and $S, T \in \mathscr{M}$. Why is $S \cup T \in \mathscr{M}$? +Back: The additive property of $a$ states it is. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $a$ is an area function and $S, T \in \mathscr{M}$. Why is $S \cap T \in \mathscr{M}$? +Back: The additive property of $a$ states it is. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $a$ is an area function and $S, T \in \mathscr{M}$. What does $a(S \cup T)$ evaluate to? +Back: $a(S) + a(T) - a(S \cap T)$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +The additive property of area uses what combinatorial concept? +Back: The principle of inclusion/exclusion. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +### Difference Property + +If $S, T \in \mathscr{M}$ such that $S \subseteq T$, then $T - S \in \mathscr{M}$ and $$a(T - S) = a(T) - a(S).$$ + +%%ANKI +Basic +Suppose $S, T \in \mathscr{M}$. What set(s) does the difference property of area state are also in $\mathscr{M}$? +Back: N/A. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $S, T \in \mathscr{M}$ such that $S \subseteq T$. What set(s) does the difference property of area state are also in $\mathscr{M}$? +Back: $T - S$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $S, T \in \mathscr{M}$ such that $T \subseteq S$. What set(s) does the difference property of area state are also in $\mathscr{M}$? +Back: $S - T$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $a$ is an area function and $S, T \in \mathscr{M}$ s.t. $S \subseteq T$. Why is $T - S \in \mathscr{M}$? +Back: The difference property of $a$ states it is. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $a$ is an area function and $S, T \in \mathscr{M}$ s.t. $S \subseteq T$. What does $a(T - S)$ evaluate to? +Back: $a(T) - a(S)$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +### Invariance Under Congruence + +If $S \in \mathscr{M}$ and $T$ is congruent to $S$, then $T \in \mathscr{M}$ and $a(S) = a(T)$. + +%%ANKI +Basic +What does the invariance of congruence property of area state? +Back: If $S \in \mathscr{M}$ and $T$ is congruent to $S$, then $T \in \mathscr{M}$ and $a(S) = a(T)$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $S \in \mathscr{M}$ and $T$ is congruent to $S$. What set(s) does the invariance of congruence property of area state are also in $\mathscr{M}$? +Back: $T$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $S \in \mathscr{M}$ and $T$ is congruent to $S$. What does $a(T)$ evaluate to? +Back: $a(S)$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +### Choice of Scale + +Every rectangle $R$ is in $\mathscr{M}$. If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$. + +%%ANKI +Basic +What shape is the choice of scale property of area concerned with? +Back: Rectangles. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What sets does the choice of scale property of area state are also in $\mathscr{M}$? +Back: All rectangles. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $R$ is a rectangle. What property of area claims $R$ is measurable? +Back: Choice of scale. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose $R$ is a rectangle. What does $a(R)$ evaluate to? +Back: If $R$ has edges of length $h$ and $k$, $a(R) = hk$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What is the area of a line segment? +Back: $0$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +The line segment is considered a special case of what other shape? +Back: A rectangle. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +How does a rectangle relate to a line segment? +Back: A line segment is a rectangle with one dimension equal to zero. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +What is the area of a point? +Back: $0$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +The point is considered a special case of what other shape? +Back: A rectangle. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +How does a rectangle relate to a point? +Back: A point is a rectangle with both dimensions equal to zero. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +### Exhaustion Property + +Let $Q$ be a set. If there exists exactly one $c$ such that $a(S) \leq c \leq a(T)$ for all step regions $S$ and $T$ satisfying $S \subseteq Q \subseteq T$, then $Q \in \mathscr{M}$ and $a(Q) = c$. + +%%ANKI +Cloze +Let $Q$ be a set. The {exhaustion} property of area states that if there exists {exactly one} $c$ such that {$a(S) \leq c \leq a(T)$} for all {step regions} $S$ and $T$ satisfying {$S \subseteq Q \subseteq T$}, then {$Q \in \mathscr{M}$} and {$a(Q) = c$}. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +The exhaustion property of area considers sets bounded by what? +Back: Step regions. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +*Why* does the exhaustion property of area require existence of exactly one satisfying real number? +Back: Area is a function, i.e. single-valued. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Which axiom of area is typically used to prove ordinate sets are measurable? +Back: The exhaustion property. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +## Bibliography + +* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). \ No newline at end of file diff --git a/notes/geometry/cartesian.md b/notes/geometry/cartesian.md index e451341..3002bfd 100644 --- a/notes/geometry/cartesian.md +++ b/notes/geometry/cartesian.md @@ -24,6 +24,30 @@ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Int END%% +%%ANKI +Basic +What is an ordinate set? +Back: A set bounded by the $x$-axis and the graph of a nonnegative function. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +An ordinate set is bounded below by what? +Back: The $x$-axis, i.e. $y = 0$. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +An ordinate set is bounded above by what? +Back: The graph of a nonnegative function. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + %%ANKI Cloze The {origin} of a Cartesian coordinate system has coordinates $\langle 0, 0 \rangle$. diff --git a/notes/geometry/index.md b/notes/geometry/index.md index 83e7e7d..1f37dfa 100644 --- a/notes/geometry/index.md +++ b/notes/geometry/index.md @@ -1,3 +1,27 @@ --- title: Geometry --- + +## Overview + +Two sets are **congruent** if their points can be put in one-to-one correspondence in such a way that distances are preserved. + +%%ANKI +Basic +Suppose sets $P$ and $Q$ are congruent. What does this imply the existence of? +Back: A bijection between $P$ and $Q$ that preserves distances between points. +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +%%ANKI +Basic +Suppose sets $P$ and $Q$ are congruent and $f$ is the corresponding bijection. What FOL proposition follows? +Back: $\forall p_1, p_2 \in P, \lvert p_1 - p_2 \rvert = \lvert f(p_1) - f(p_2) \rvert$ +Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). + +END%% + +## Bibliography + +* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). \ No newline at end of file