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Axioms of area.

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Joshua Potter 2024-11-23 10:03:08 -07:00
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---
title: "2024-11-23"
---
- [ ] Anki Flashcards
- [x] KoL
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- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)

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---
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).
<!--ID: 1732381333289-->
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).
<!--ID: 1732381333310-->
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).
<!--ID: 1732381333313-->
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).
<!--ID: 1732381333316-->
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).
<!--ID: 1732381333319-->
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).
<!--ID: 1732381333321-->
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).
<!--ID: 1732381333324-->
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).
<!--ID: 1732381333327-->
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).
<!--ID: 1732381333329-->
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).
<!--ID: 1732381333332-->
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).
<!--ID: 1732381333334-->
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).
<!--ID: 1732381333337-->
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).
<!--ID: 1732381333340-->
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).
<!--ID: 1732381333343-->
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).
<!--ID: 1732381333346-->
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).
<!--ID: 1732381333349-->
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).
<!--ID: 1732381333353-->
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).
<!--ID: 1732381333357-->
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).
<!--ID: 1732381333361-->
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).
<!--ID: 1732381333365-->
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).
<!--ID: 1732381333368-->
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).
<!--ID: 1732381333372-->
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).
<!--ID: 1732381333376-->
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).
<!--ID: 1732381333380-->
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).
<!--ID: 1732381333384-->
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).
<!--ID: 1732381333388-->
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).
<!--ID: 1732381333391-->
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).
<!--ID: 1732381333395-->
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).
<!--ID: 1732381333399-->
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).
<!--ID: 1732381333403-->
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).
<!--ID: 1732381333409-->
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).
<!--ID: 1732381333414-->
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).
<!--ID: 1732381333419-->
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).
<!--ID: 1732381333427-->
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).
<!--ID: 1732381333433-->
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).
<!--ID: 1732381333438-->
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).
<!--ID: 1732381333444-->
END%%
## Bibliography
* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).

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<!--ID: 1731184865791-->
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).
<!--ID: 1732381333459-->
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).
<!--ID: 1732381333464-->
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).
<!--ID: 1732381333469-->
END%%
%%ANKI
Cloze
The {origin} of a Cartesian coordinate system has coordinates $\langle 0, 0 \rangle$.

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---
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).
<!--ID: 1732381333449-->
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).
<!--ID: 1732381333454-->
END%%
## Bibliography
* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).