From 50d6b13574b05c01f87054aee5a1b92b608439de Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Wed, 10 May 2023 20:27:46 -0600 Subject: [PATCH] Remove no longer needed `hyperlabel` command. --- Bookshelf/Apostol/Chapter_1_07.tex | 24 +++++++++++------------ Bookshelf/Apostol/Chapter_1_11.tex | 24 +++++++++++------------ Bookshelf/Apostol/Chapter_I_03.tex | 30 ++++++++++++++--------------- Bookshelf/Enderton/Chapter_0.tex | 2 +- Common/Real/Geometry/Area.tex | 10 +++++----- Common/Real/Sequence/Arithmetic.tex | 2 +- Common/Real/Sequence/Geometric.tex | 2 +- preamble.tex | 3 --- 8 files changed, 47 insertions(+), 50 deletions(-) diff --git a/Bookshelf/Apostol/Chapter_1_07.tex b/Bookshelf/Apostol/Chapter_1_07.tex index d265578..fb539e3 100644 --- a/Bookshelf/Apostol/Chapter_1_07.tex +++ b/Bookshelf/Apostol/Chapter_1_07.tex @@ -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 $A$. diff --git a/Bookshelf/Apostol/Chapter_1_11.tex b/Bookshelf/Apostol/Chapter_1_11.tex index 31f27e9..dc8cff4 100644 --- a/Bookshelf/Apostol/Chapter_1_11.tex +++ b/Bookshelf/Apostol/Chapter_1_11.tex @@ -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 diff --git a/Bookshelf/Apostol/Chapter_I_03.tex b/Bookshelf/Apostol/Chapter_I_03.tex index 20f2dc0..c4838b7 100644 --- a/Bookshelf/Apostol/Chapter_I_03.tex +++ b/Bookshelf/Apostol/Chapter_I_03.tex @@ -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 $T$ diff --git a/Bookshelf/Enderton/Chapter_0.tex b/Bookshelf/Enderton/Chapter_0.tex index 24f8a4e..2e16866 100644 --- a/Bookshelf/Enderton/Chapter_0.tex +++ b/Bookshelf/Enderton/Chapter_0.tex @@ -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 $\langle x_1, \ldots, x_m \rangle = \langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$. diff --git a/Common/Real/Geometry/Area.tex b/Common/Real/Geometry/Area.tex index 138324c..31ab25d 100644 --- a/Common/Real/Geometry/Area.tex +++ b/Common/Real/Geometry/Area.tex @@ -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 diff --git a/Common/Real/Sequence/Arithmetic.tex b/Common/Real/Sequence/Arithmetic.tex index 6d00dca..c7d58c2 100644 --- a/Common/Real/Sequence/Arithmetic.tex +++ b/Common/Real/Sequence/Arithmetic.tex @@ -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 $n \in \mathbb{N}$, diff --git a/Common/Real/Sequence/Geometric.tex b/Common/Real/Sequence/Geometric.tex index 9a5611e..5cafa53 100644 --- a/Common/Real/Sequence/Geometric.tex +++ b/Common/Real/Sequence/Geometric.tex @@ -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 $n \in \mathbb{N}$, diff --git a/preamble.tex b/preamble.tex index cde5b39..30dab7f 100644 --- a/preamble.tex +++ b/preamble.tex @@ -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