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Clean-up section colors.

finite-set-exercises
Joshua Potter 2023-05-13 09:32:13 -06:00
parent 708295f53f
commit 0744023a9d
2 changed files with 12 additions and 18 deletions
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24
Bookshelf/Apostol.tex
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@ -569,7 +569,7 @@ If there is one and only one number $c$ which satisfies the inequalities
Prove that each of the following sets is measurable and has zero area: Prove that each of the following sets is measurable and has zero area:
\subsection{\unverified{Exercise 1.7.1a}}% \subsection{\partial{Exercise 1.7.1a}}%
\label{sub:exercise-1.7.1a} \label{sub:exercise-1.7.1a}
A set consisting of a single point. A set consisting of a single point.
@ -585,7 +585,7 @@ A set consisting of a single point.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 1.7.1b}}% \subsection{\partial{Exercise 1.7.1b}}%
\label{sub:exercise-1.7.1b} \label{sub:exercise-1.7.1b}
A set consisting of a finite number of points in a plane. A set consisting of a finite number of points in a plane.
@ -644,7 +644,7 @@ A set consisting of a finite number of points in a plane.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 1.7.1c}}% \subsection{\partial{Exercise 1.7.1c}}%
\label{sub:exercise-1.7.1c} \label{sub:exercise-1.7.1c}
The union of a finite collection of line segments in a plane. The union of a finite collection of line segments in a plane.
@ -707,7 +707,7 @@ The union of a finite collection of line segments in a plane.
\end{proof} \end{proof}
\section{\unverified{Exercise 1.7.2}}% \section{\partial{Exercise 1.7.2}}%
\label{sec:exercise-1.7.2} \label{sec:exercise-1.7.2}
Every right triangular region is measurable because it can be obtained as the Every right triangular region is measurable because it can be obtained as the
@ -758,7 +758,7 @@ Prove that every triangular region is measurable and that its area is one half
\end{proof} \end{proof}
\section{\unverified{Exercise 1.7.3}}% \section{\partial{Exercise 1.7.3}}%
\label{sec:exercise-1.7.3} \label{sec:exercise-1.7.3}
Prove that every trapezoid and every parallelogram is measurable and derive the Prove that every trapezoid and every parallelogram is measurable and derive the
@ -871,7 +871,7 @@ Let $P$ be a polygon whose vertices are lattice points.
The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of
lattice points inside the polygon and $B$ denotes the number on the boundary. lattice points inside the polygon and $B$ denotes the number on the boundary.
\subsection{\unverified{Exercise 1.7.4a}}% \subsection{\partial{Exercise 1.7.4a}}%
\label{sub:exercise-1.7.4a} \label{sub:exercise-1.7.4a}
Prove that the formula is valid for rectangles with sides parallel to the Prove that the formula is valid for rectangles with sides parallel to the
@ -899,7 +899,7 @@ Prove that the formula is valid for rectangles with sides parallel to the
\end{proof} \end{proof}
\subsection{\unverified{Exercise 1.7.4b}}% \subsection{\partial{Exercise 1.7.4b}}%
\label{sub:exercise-1.7.4b} \label{sub:exercise-1.7.4b}
Prove that the formula is valid for right triangles and parallelograms. Prove that the formula is valid for right triangles and parallelograms.
@ -948,7 +948,7 @@ Prove that the formula is valid for right triangles and parallelograms.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 1.7.4c}}% \subsection{\partial{Exercise 1.7.4c}}%
\label{sub:exercise-1.7.4c} \label{sub:exercise-1.7.4c}
Use induction on the number of edges to construct a proof for general polygons. Use induction on the number of edges to construct a proof for general polygons.
@ -1014,7 +1014,7 @@ Use induction on the number of edges to construct a proof for general polygons.
\end{proof} \end{proof}
\section{\unverified{Exercise 1.7.5}}% \section{\partial{Exercise 1.7.5}}%
\label{sec:exercise-1.7.5} \label{sec:exercise-1.7.5}
Prove that a triangle whose vertices are lattice points cannot be equilateral. Prove that a triangle whose vertices are lattice points cannot be equilateral.
@ -1050,7 +1050,7 @@ ways, using Exercises 2 and 4.]
\end{proof} \end{proof}
\section{\unverified{Exercise 1.7.6}}% \section{\partial{Exercise 1.7.6}}%
\label{sec:exercise-1.7.6} \label{sec:exercise-1.7.6}
Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all
@ -1385,7 +1385,7 @@ State and prove such a generalization.
\end{proof} \end{proof}
\section{\unverified{Exercise 1.11.6}}% \section{\partial{Exercise 1.11.6}}%
\label{sec:exercise-1.11.6} \label{sec:exercise-1.11.6}
Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
@ -1438,7 +1438,7 @@ When $b = 1$, the sum on the left is understood to be $0$.
\note{When $b = 1$, the proofs of (a) and (b) are trivial. We continue under the \note{When $b = 1$, the proofs of (a) and (b) are trivial. We continue under the
assumption $b > 1$.} assumption $b > 1$.}
\subsection{\unverified{Exercise 1.11.7a}}% \subsection{\partial{Exercise 1.11.7a}}%
\label{sub:exercise-1.11.7a} \label{sub:exercise-1.11.7a}
Derive this result by a geometric argument, counting lattice points in a right Derive this result by a geometric argument, counting lattice points in a right

6
Bookshelf/Enderton.tex
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@ -27,10 +27,4 @@ Assume that $\langle x_1, \ldots, x_m \rangle =
\langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$. \langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$.
Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$. Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$.
\begin{proof}
TODO
\end{proof}
\end{document} \end{document}