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| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Area | Obsidian::STEM | geometry::area |
|
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).
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%%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).
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%%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).
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%%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).
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%%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).
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%%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).
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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).
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%%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).
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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.
%%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).
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%%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).
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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).
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This property lets us prove \varnothing is measurable with the expected area: $a(\varnothing) = a(T - T) = a(T) - a(T) = 0$
%%ANKI Basic Is the empty set considered measurable? Back: Yes. Reference: 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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%%ANKI
Basic
Which axiom of area is used to prove \varnothing is measurable?
Back: The difference property.
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 a(\varnothing) evaluate to?
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
What chain of equalities is used to prove that a(\varnothing) = 0?
Back: For any T \in \mathscr{M}, a(\varnothing) = a(T - T) = a(T) - a(T) = 0.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
This property also lets us state the monotone property: \forall S, T \in \mathscr{M}, S \subseteq T \Rightarrow a(S) \leq a(T)
%%ANKI
Basic
What does the monotone property of area state in FOL?
Back: \forall S, T \in \mathscr{M}, S \subseteq T \Rightarrow a(S) \leq 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
What name is given to the following property? \forall S, T \in \mathscr{M}, S \subseteq T \Rightarrow a(S) \leq a(T)
Back: The monotone property.
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(s) of area are used to prove the monotone property? Back: The nonnegative and difference properties. Reference: 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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%%ANKI
Basic
Let S, T \in \mathscr{M} such that S \subseteq T. Why is the difference property needed to show a(T - S) \geq 0?
Back: It states T - S is measurable.
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
Let S, T \in \mathscr{M} such that S \subseteq T. Why is the difference property needed to show a(S - T) \geq 0?
Back: N/A. The difference property says nothing about 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
Let S, T \in \mathscr{M} such that T \subseteq S. Why is the difference property needed to show a(S - T) \geq 0?
Back: It states S - T is measurable.
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
Let S, T \in \mathscr{M} such that S \subseteq T. Why is the nonnegative property needed to show a(T - S) \geq 0?
Back: It states the area of any measurable set is \geq 0.
Reference: 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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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).
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%%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).
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%%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).
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%%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).
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%%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).
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%%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).
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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:
If there exists {exactly one} c \in \mathbb{R} 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).
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%%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).
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%%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).
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%%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).
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Bibliography
- Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).