Remove no longer needed `hyperlabel` command.

finite-set-exercises
Joshua Potter 2023-05-10 20:27:46 -06:00
parent 53a0bd1ebc
commit 50d6b13574
8 changed files with 47 additions and 50 deletions

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@ -18,12 +18,12 @@ The properties of area in this set of exercises are to be deduced from the
axioms for area stated in the foregoing section.
\section*{Exercise 1}%
\hyperlabel{sec:exercise-1}%
\label{sec:exercise-1}
Prove that each of the following sets is measurable and has zero area:
\subsection*{\proceeding{Exercise 1a}}%
\hyperlabel{sub:exercise-1a}%
\label{sub:exercise-1a}
A set consisting of a single point.
@ -39,7 +39,7 @@ A set consisting of a single point.
\end{proof}
\subsection*{\proceeding{Exercise 1b}}%
\hyperlabel{sub:exercise-1b}%
\label{sub:exercise-1b}
A set consisting of a finite number of points in a plane.
@ -98,7 +98,7 @@ A set consisting of a finite number of points in a plane.
\end{proof}
\subsection*{\proceeding{Exercise 1c}}%
\hyperlabel{sub:exercise-1c}%
\label{sub:exercise-1c}
The union of a finite collection of line segments in a plane.
@ -161,7 +161,7 @@ The union of a finite collection of line segments in a plane.
\end{proof}
\section*{\unverified{Exercise 2}}%
\hyperlabel{sec:exercise-2}%
\label{sec:exercise-2}
Every right triangular region is measurable because it can be obtained as the
intersection of two rectangles.
@ -212,7 +212,7 @@ Prove that every triangular region is measurable and that its area is one half
\end{proof}
\section*{\unverified{Exercise 3}}%
\hyperlabel{sec:exercise-3}%
\label{sec:exercise-3}
Prove that every trapezoid and every parallelogram is measurable and derive the
usual formulas for their areas.
@ -319,14 +319,14 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
\end{proof}
\section*{Exercise 4}%
\hyperlabel{sec:exercise-4}%
\label{sec:exercise-4}
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
lattice points inside the polygon and $B$ denotes the number on the boundary.
\subsection*{\unverified{Exercise 4a}}%
\hyperlabel{sub:exercise-4a}%
\label{sub:exercise-4a}
Prove that the formula is valid for rectangles with sides parallel to the
coordinate axes.
@ -354,7 +354,7 @@ Prove that the formula is valid for rectangles with sides parallel to the
\end{proof}
\subsection*{\unverified{Exercise 4b}}%
\hyperlabel{sub:exercise-4b}%
\label{sub:exercise-4b}
Prove that the formula is valid for right triangles and parallelograms.
@ -408,7 +408,7 @@ Prove that the formula is valid for right triangles and parallelograms.
\end{proof}
\subsection*{\unverified{Exercise 4c}}%
\hyperlabel{sub:exercise-4c}%
\label{sub:exercise-4c}
Use induction on the number of edges to construct a proof for general polygons.
@ -474,7 +474,7 @@ Use induction on the number of edges to construct a proof for general polygons.
\end{proof}
\section*{\unverified{Exercise 5}}%
\hyperlabel{sec:exercise-5}%
\label{sec:exercise-5}
Prove that a triangle whose vertices are lattice points cannot be equilateral.
@ -510,7 +510,7 @@ ways, using Exercises 2 and 4.]
\end{proof}
\section*{\unverified{Exercise 6}}%
\hyperlabel{sec:exercise-6}%
\label{sec:exercise-6}
Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all
subsets of $A$.

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@ -11,12 +11,12 @@
\header{Exercises 1.11}{Tom M. Apostol}
\section*{Exercise 4}%
\hyperlabel{sec:exercise-4}%
\label{sec:exercise-4}
Prove that the greatest-integer function has the properties indicated:
\subsection*{\proceeding{Exercise 4a}}%
\hyperlabel{sub:exercise-4a}%
\label{sub:exercise-4a}
$\floor{x + n} = \floor{x} + n$ for every integer $n$.
@ -27,7 +27,7 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$.
\end{proof}
\subsection*{\proceeding{Exercise 4b}}%
\hyperlabel{sub:exercise-4b}%
\label{sub:exercise-4b}
$\floor{-x} =
\begin{cases}
@ -47,7 +47,7 @@ $\floor{-x} =
\end{proof}
\subsection*{\proceeding{Exercise 4c}}%
\hyperlabel{sub:exercise-4c}%
\label{sub:exercise-4c}
$\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
@ -58,7 +58,7 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
\end{proof}
\subsection*{\proceeding{Exercise 4d}}%
\hyperlabel{sub:exercise-4d}%
\label{sub:exercise-4d}
$\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
@ -69,7 +69,7 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
\end{proof}
\subsection*{\proceeding{Exercise 4e}}%
\hyperlabel{sub:exercise-4e}%
\label{sub:exercise-4e}
$\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
@ -80,7 +80,7 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
\end{proof}
\section*{\proceeding{Exercise 5}}%
\hyperlabel{sec:exercise-5}%
\label{sec:exercise-5}
The formulas in Exercises 4(d) and 4(e) suggest a generalization for
$\floor{nx}$.
@ -169,7 +169,7 @@ State and prove such a generalization.
\end{proof}
\section*{\unverified{Exercise 6}}%
\hyperlabel{sec:exercise-6}%
\label{sec:exercise-6}
Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
integers.
@ -193,14 +193,14 @@ Prove that the number of lattice points in $S$ is equal to the sum
\end{proof}
\section*{Exercise 7}%
\hyperlabel{sec:exercise-7}%
\label{sec:exercise-7}
If $a$ and $b$ are positive integers with no common factor, we have the formula
$$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$
When $b = 1$, the sum on the left is understood to be $0$.
\subsection*{\unverified{Exercise 7a}}%
\hyperlabel{sub:exercise-7a}%
\label{sub:exercise-7a}
Derive this result by a geometric argument, counting lattice points in a right
triangle.
@ -212,7 +212,7 @@ Derive this result by a geometric argument, counting lattice points in a right
\end{proof}
\subsection*{\proceeding{Exercise 7b}}%
\hyperlabel{sub:exercise-7b}%
\label{sub:exercise-7b}
Derive the result analytically as follows:
By changing the index of summation, note that
@ -226,7 +226,7 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
\end{proof}
\section*{\unverified{Exercise 8}}%
\hyperlabel{sec:exercise-8}%
\label{sec:exercise-8}
Let $S$ be a set of points on the real line.
The \textit{characteristic function} of $S$ is, by definition, the function

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@ -11,7 +11,7 @@
\header{A Set of Axioms for the Real-Number System}{Tom M. Apostol}
\section*{\verified{Lemma 1}}%
\hyperlabel{sec:lemma-1}%
\label{sec:lemma-1}
Nonempty set $S$ has supremum $L$ if and only if set $-S$ has infimum $-L$.
@ -31,7 +31,7 @@ Nonempty set $S$ has supremum $L$ if and only if set $-S$ has infimum $-L$.
\end{proof}
\section*{\verified{Theorem I.27}}%
\hyperlabel{sec:theorem-i.27}%
\label{sec:theorem-i.27}
Every nonempty set $S$ that is bounded below has a greatest lower bound; that
is, there is a real number $L$ such that $L = \inf{S}$.
@ -51,7 +51,7 @@ Every nonempty set $S$ that is bounded below has a greatest lower bound; that
\end{proof}
\section*{\verified{Theorem I.29}}%
\hyperlabel{sec:theorem-i.29}
\label{sec:theorem-i.29}
For every real $x$ there exists a positive integer $n$ such that $n > x$.
@ -71,7 +71,7 @@ For every real $x$ there exists a positive integer $n$ such that $n > x$.
\end{proof}
\section*{\verified{Theorem I.30}}%
\hyperlabel{sec:theorem-i.30}%
\label{sec:theorem-i.30}
If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
integer $n$ such that $nx > y$.
@ -92,7 +92,7 @@ If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
\end{proof}
\section*{\verified{Theorem I.31}}%
\hyperlabel{sec:theorem-i.31}%
\label{sec:theorem-i.31}
If three real numbers $a$, $x$, and $y$ satisfy the inequalities
$$a \leq x \leq a + \frac{y}{n}$$ for every integer $n \geq 1$, then $x = a$.
@ -135,7 +135,7 @@ If three real numbers $a$, $x$, and $y$ satisfy the inequalities
\end{proof}
\section*{\verified{Lemma 2}}%
\hyperlabel{sec:lemma-2}%
\label{sec:lemma-2}
If three real numbers $a$, $x$, and $y$ satisfy the inequalities
$$a - y / n \leq x \leq a$$ for every integer $n \geq 1$, then $x = a$.
@ -178,12 +178,12 @@ If three real numbers $a$, $x$, and $y$ satisfy the inequalities
\end{proof}
\section*{Theorem I.32}%
\hyperlabel{sec:theorem-i.32}%
\label{sec:theorem-i.32}
Let $h$ be a given positive number and let $S$ be a set of real numbers.
\subsection*{\verified{Theorem I.32a}}%
\hyperlabel{sub:theorem-i.32a}%
\label{sub:theorem-i.32a}
If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup{S} - h$.
@ -205,7 +205,7 @@ If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup{S} - h$.
\end{proof}
\subsection*{\verified{Theorem I.32b}}%
\hyperlabel{sub:theorem-i.32b}%
\label{sub:theorem-i.32b}
If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf{S} + h$.
@ -227,7 +227,7 @@ If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf{S} + h$.
\end{proof}
\section*{Theorem I.33}%
\hyperlabel{sec:theorem-i.33}%
\label{sec:theorem-i.33}
Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
$$C = \{a + b : a \in A, b \in B\}.$$
@ -235,7 +235,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
\note{This is known as the "Additive Property."}
\subsection*{\verified{Theorem I.33a}}%
\hyperlabel{sub:theorem-i.33a}%
\label{sub:theorem-i.33a}
If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
$$\sup{C} = \sup{A} + \sup{B}.$$
@ -250,7 +250,7 @@ If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
$\sup{A} + \sup{B}$ is the \textit{least} upper bound of $C$.
\paragraph{(i)}%
\hyperlabel{par:theorem-i.33a-i}%
\label{par:theorem-i.33a-i}
Let $x \in C$.
By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such
@ -303,7 +303,7 @@ If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
\end{proof}
\subsection*{\verified{Theorem I.33b}}%
\hyperlabel{sub:theorem-i.33b}%
\label{sub:theorem-i.33b}
If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
$$\inf{C} = \inf{A} + \inf{B}.$$
@ -318,7 +318,7 @@ If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
$\inf{A} + \inf{B}$ is the \textit{greatest} lower bound of $C$.
\paragraph{(i)}%
\hyperlabel{par:theorem-i.33b-i}%
\label{par:theorem-i.33b-i}
Let $x \in C$.
By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such
@ -371,7 +371,7 @@ If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
\end{proof}
\section*{\verified{Theorem I.34}}%
\hyperlabel{sec:theorem-i.34}%
\label{sec:theorem-i.34}
Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that $$s \leq t$$
for every $s$ in $S$ and every $t$ in $T$. Then $S$ has a supremum, and $T$

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@ -11,7 +11,7 @@
\header{Useful Facts About Sets}{Herbert B. Enderton}
\section*{\proceeding{Lemma 0A}}%
\hyperlabel{sec:lemma-0a}%
\label{sec:lemma-0a}
Assume that $\langle x_1, \ldots, x_m \rangle =
\langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$.

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@ -13,7 +13,7 @@ We assume there exists a class $\mathscr{M}$ of measurable sets in the plane and
properties:
\section*{\defined{Nonnegative Property}}%
\hyperlabel{sec:nonnegative-property}%
\label{sec:nonnegative-property}
For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
@ -24,7 +24,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
\end{axiom}
\section*{\defined{Additive Property}}%
\hyperlabel{sec:additive-property}%
\label{sec:additive-property}
If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
$\mathscr{M}$, and we have $a(S \cup T) = a(S) + a(T) - a(S \cap T)$.
@ -36,7 +36,7 @@ If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
\end{axiom}
\section*{\defined{Difference Property}}%
\hyperlabel{sec:difference-property}%
\label{sec:difference-property}
If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
$\mathscr{M}$, and we have $a(T - S) = a(T) - a(S)$.
@ -48,7 +48,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
\end{axiom}
\section*{\defined{Invariance Under Congruence}}%
\hyperlabel{sec:invariance-under-congruence}%
\label{sec:invariance-under-congruence}
If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
also in $\mathscr{M}$ and we have $a(S) = a(T)$.
@ -72,7 +72,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
\end{axiom}
\section*{\proceeding{Exhaustion Property}}%
\hyperlabel{sec:exhaustion-property}%
\label{sec:exhaustion-property}
Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so
that

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@ -9,7 +9,7 @@
\begin{document}
\section{\proceeding{Sum of Arithmetic Series}}%
\hyperlabel{sec:sum-arithmetic-series}%
\label{sec:sum-arithmetic-series}
Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
Then for some $n \in \mathbb{N}$,

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@ -9,7 +9,7 @@
\begin{document}
\section{\proceeding{Sum of Geometric Series}}%
\hyperlabel{sec:sum-geometric-series}%
\label{sec:sum-geometric-series}
Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
Then for some $n \in \mathbb{N}$,

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@ -16,9 +16,6 @@
\hypersetup{colorlinks=true, urlcolor=blue}
\newcommand{\leanref}[2]{\color{blue}$\pmb{\exists}\;{-}\;$\href{#1}{#2}}
\newcommand{\hyperlabel}[1]{%
\label{#1}%
\hypertarget{#1}{}}
% ========================================
% Environments