99 lines
3.2 KiB
Plaintext
99 lines
3.2 KiB
Plaintext
/-
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Exercises 1.7
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-/
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import OneVariableCalculus.Real.Geometry.Area
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import OneVariableCalculus.Real.Geometry.Rectangle
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open Real
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open Real.Geometry.Area
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-- ========================================
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-- The properties of area in this set of exercises are to be deduced from the
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-- axioms for area stated in the foregoing section.
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-- ========================================
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-- ----------------------------------------
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-- Exercise 1
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--
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-- Prove that each of the following sets is measurable and has zero area:
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-- (a) A set consisting of a single point.
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-- (b) A set consisting of a finite number of points in a plane.
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-- (c) The union of a finite collection of line segments in a plane.
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-- ----------------------------------------
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example (p : Point)
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: p.set_def ∈ 𝓜
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∧ ((h : p.set_def ∈ 𝓜) → area h = 0) :=
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sorry
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example (S : Set Point) (hf : Set.Finite S)
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: (⋃ p ∈ S, p.set_def) ∈ 𝓜
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∧ ((h : (⋃ p ∈ S, p.set_def) ∈ 𝓜) → area h = 0) :=
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sorry
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example (S : Set LineSegment) (hf : Set.Finite S)
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: (⋃ s ∈ S, s.set_def) ∈ 𝓜
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∧ ((h : (⋃ s ∈ S, s.set_def) ∈ 𝓜) → area h = 0) :=
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sorry
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-- ----------------------------------------
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-- Exercise 2
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--
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-- Every right triangular region is measurable because it can be obtained as the
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-- intersection of two rectangles. Prove that every triangular region is
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-- measurable and that its area is one half the product of its base and
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-- altitude.
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-- ----------------------------------------
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-- # TODO
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-- ----------------------------------------
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-- Exercise 3
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--
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-- Prove that every trapezoid and every parallelogram is measurable and derive
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-- the usual formulas for their areas.
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-- ----------------------------------------
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-- # TODO
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-- ----------------------------------------
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-- Exercise 4
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--
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-- A point `(x, y)` in the plane is called a *lattice point* if both coordinates
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-- `x` and `y` are integers. Let `P` be a polygon whose vertices are lattice
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-- points inside the polygon and `B` denotes the number on the boundary.
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--
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-- (a) Prove that the formula is valid for rectangles with sides parallel to the
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-- coordinate axes.
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-- (b) Prove that the formula is valid for right triangles and parallelograms.
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-- (c) Use induction on the number of edges to construct a proof for general
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-- polygons.
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-- ----------------------------------------
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-- # TODO
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-- ----------------------------------------
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-- Exercise 5
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--
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-- Prove that a triangle whose vertices are lattice points cannot be
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-- equilateral.
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--
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-- [Hint: Assume there is such a triangle and compute its area in two ways,
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-- using exercises 2 and 4.]
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-- ----------------------------------------
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-- # TODO
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-- ----------------------------------------
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-- Exercise 6
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--
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-- Let `A = {1, 2, 3, 4, 5}`, and let `𝓜` denote the class of all subsets of
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-- `A`. (There are `32` altogether, counting `A` itself and the empty set `∅`.)
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-- For each set `S` in `𝓜`, let `n(S)` denote the number of distinct elements in
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-- `S`. If `S = {1, 2, 3, 4}` and `T = {3, 4, 5}`, compute `n(S ∪ T)`,
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-- `n(S ∩ T)`, `n(S - T)`, and `n(T - S)`. Prove that the set function `n`
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-- satisfies the first three axioms for area.
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-- ----------------------------------------
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-- # TODO
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