Clean-up section colors.
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@ -569,7 +569,7 @@ If there is one and only one number $c$ which satisfies the inequalities
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Prove that each of the following sets is measurable and has zero area:
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\subsection{\unverified{Exercise 1.7.1a}}%
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\subsection{\partial{Exercise 1.7.1a}}%
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\label{sub:exercise-1.7.1a}
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A set consisting of a single point.
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@ -585,7 +585,7 @@ A set consisting of a single point.
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\end{proof}
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\subsection{\unverified{Exercise 1.7.1b}}%
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\subsection{\partial{Exercise 1.7.1b}}%
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\label{sub:exercise-1.7.1b}
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A set consisting of a finite number of points in a plane.
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@ -644,7 +644,7 @@ A set consisting of a finite number of points in a plane.
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\end{proof}
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\subsection{\unverified{Exercise 1.7.1c}}%
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\subsection{\partial{Exercise 1.7.1c}}%
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\label{sub:exercise-1.7.1c}
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The union of a finite collection of line segments in a plane.
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@ -707,7 +707,7 @@ The union of a finite collection of line segments in a plane.
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\end{proof}
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\section{\unverified{Exercise 1.7.2}}%
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\section{\partial{Exercise 1.7.2}}%
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\label{sec:exercise-1.7.2}
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Every right triangular region is measurable because it can be obtained as the
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@ -758,7 +758,7 @@ Prove that every triangular region is measurable and that its area is one half
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\end{proof}
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\section{\unverified{Exercise 1.7.3}}%
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\section{\partial{Exercise 1.7.3}}%
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\label{sec:exercise-1.7.3}
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Prove that every trapezoid and every parallelogram is measurable and derive the
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@ -871,7 +871,7 @@ Let $P$ be a polygon whose vertices are lattice points.
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The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of
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lattice points inside the polygon and $B$ denotes the number on the boundary.
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\subsection{\unverified{Exercise 1.7.4a}}%
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\subsection{\partial{Exercise 1.7.4a}}%
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\label{sub:exercise-1.7.4a}
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Prove that the formula is valid for rectangles with sides parallel to the
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@ -899,7 +899,7 @@ Prove that the formula is valid for rectangles with sides parallel to the
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\end{proof}
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\subsection{\unverified{Exercise 1.7.4b}}%
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\subsection{\partial{Exercise 1.7.4b}}%
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\label{sub:exercise-1.7.4b}
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Prove that the formula is valid for right triangles and parallelograms.
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@ -948,7 +948,7 @@ Prove that the formula is valid for right triangles and parallelograms.
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\end{proof}
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\subsection{\unverified{Exercise 1.7.4c}}%
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\subsection{\partial{Exercise 1.7.4c}}%
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\label{sub:exercise-1.7.4c}
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Use induction on the number of edges to construct a proof for general polygons.
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@ -1014,7 +1014,7 @@ Use induction on the number of edges to construct a proof for general polygons.
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\end{proof}
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\section{\unverified{Exercise 1.7.5}}%
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\section{\partial{Exercise 1.7.5}}%
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\label{sec:exercise-1.7.5}
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Prove that a triangle whose vertices are lattice points cannot be equilateral.
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@ -1050,7 +1050,7 @@ ways, using Exercises 2 and 4.]
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\end{proof}
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\section{\unverified{Exercise 1.7.6}}%
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\section{\partial{Exercise 1.7.6}}%
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\label{sec:exercise-1.7.6}
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Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all
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@ -1385,7 +1385,7 @@ State and prove such a generalization.
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\end{proof}
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\section{\unverified{Exercise 1.11.6}}%
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\section{\partial{Exercise 1.11.6}}%
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\label{sec:exercise-1.11.6}
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Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
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@ -1438,7 +1438,7 @@ When $b = 1$, the sum on the left is understood to be $0$.
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\note{When $b = 1$, the proofs of (a) and (b) are trivial. We continue under the
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assumption $b > 1$.}
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\subsection{\unverified{Exercise 1.11.7a}}%
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\subsection{\partial{Exercise 1.11.7a}}%
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\label{sub:exercise-1.11.7a}
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Derive this result by a geometric argument, counting lattice points in a right
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@ -27,10 +27,4 @@ Assume that $\langle x_1, \ldots, x_m \rangle =
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\langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$.
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Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$.
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\begin{proof}
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TODO
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\end{proof}
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\end{document}
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