Add prompts for Apostol, exercises 1.7.

one-variable-calculus/Apostol/Exercises/Exercises_1_7.lean

@@ -0,0 +1,80 @@
+/-
+Exercises 1.7
+-/
+
+-- ========================================
+-- The properties of area in this set of exercises are to be deduced from the
+-- axioms for area stated in the foregoing section.
+-- ========================================
+
+-- ----------------------------------------
+-- Exercise 1
+--
+-- Prove that each of the following sets is measurable and has zero area:
+-- (a) A set consisting of a single point.
+-- (b) A set consisting of a finite number of points in a plane.
+-- (c) The union of a finite collection of line segments in a plane.
+-- ----------------------------------------
+
+-- # TODO
+
+-- ----------------------------------------
+-- Exercise 2
+--
+-- Every right triangular region is measurable because it can be obtained as the
+-- intersection of two rectangles. Prove that every triangular region is
+-- measurable and that its area is one half the product of its base and
+-- altitude.
+-- ----------------------------------------
+
+-- # TODO
+
+-- ----------------------------------------
+-- Exercise 3
+--
+-- Prove that every trapezoid and every parallelogram is measurable and derive
+-- the usual formulas for their areas.
+-- ----------------------------------------
+
+-- # TODO
+
+-- ----------------------------------------
+-- Exercise 4
+--
+-- A point `(x, y)` in the plane is called a *lattice point* if both coordinates
+-- `x` and `y` are integers. Let `P` be a polygon whose vertices are lattice
+-- points inside the polygon and `B` denotes the number on the boundary.
+--
+-- (a) Prove that the formula is valid for rectangles with sides parallel to the
+-- coordinate axes.
+-- (b) Prove that the formula is valid for right triangles and parallelograms.
+-- (c) Use induction on the number of edges to construct a proof for general
+-- polygons.
+-- ----------------------------------------
+
+-- # TODO
+
+-- ----------------------------------------
+-- Exercise 5
+--
+-- Prove that a triangle whose vertices are lattice points cannot be
+-- equilateral.
+--
+-- [Hint: Assume there is such a triangle and ocmpute its area in two ways,
+-- using exercises 2 and 4.]
+-- ----------------------------------------
+
+-- # TODO
+
+-- ----------------------------------------
+-- Exercise 6
+--
+-- Let `A = {1, 2, 3, 4, 5}`, and let `𝓜` denote the class of all subsets of
+-- `A`. (There are `32` altogether, counting `A` itself and the empty set `∅`.)
+-- For each set `S` in `𝓜`, let `n(S)` denote the number of distinct elements in
+-- `S`. If `S = {1, 2, 3, 4}` and `T = {3, 4, 5}`, compute `n(S ∪ T)`,
+-- `n(S ∩ T)`, `n(S - T)`, and `n(T - S)`. Prove that the set function `n`
+-- satisfies the first three axioms for area.
+-- ----------------------------------------
+
+-- # TODO

one-variable-calculus/Apostol/Exercises/Exercises_I_3_12.lean

@@ -1,5 +1,5 @@
 /-
-I 3.12 Exercises
+Exercises I 3.12
 
 A Set of Axioms for the Real-Number System
 -/