diff --git a/Bookshelf/Apostol.tex b/Bookshelf/Apostol.tex index 201c23f..4ce1ff9 100644 --- a/Bookshelf/Apostol.tex +++ b/Bookshelf/Apostol.tex @@ -20,10 +20,10 @@ \addtocounter{chapter}{-1} \chapter{Reference}% -\label{chap:reference} +\hyperlabel{chap:reference} \section{\defined{Characteristic Function}}% -\label{ref:characteristic-function} +\hyperlabel{ref:characteristic-function} Let $S$ be a set of points on the real line. The \textbf{characteristic function} of $S$ is the function $\mathcal{X}_S$ such @@ -37,7 +37,7 @@ The \textbf{characteristic function} of $S$ is the function $\mathcal{X}_S$ such \end{definition} \section{\defined{Completeness Axiom}}% -\label{ref:completeness-axiom} +\hyperlabel{ref:completeness-axiom} Every nonempty set $S$ of real numbers which is bounded above has a supremum; that is, there is a real number $B$ such that $B = \sup{S}$. @@ -49,7 +49,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum; \end{axiom} \section{\defined{Infimum}}% -\label{ref:infimum} +\hyperlabel{ref:infimum} A number $B$ is called an \textbf{infimum} of a nonempty set $S$ if $B$ has the following two properties: @@ -66,7 +66,7 @@ Such a number $B$ is also known as the \textbf{greatest lower bound}. \end{definition} \section{\pending{Integrable}}% -\label{ref:integrable} +\hyperlabel{ref:integrable} Let $f$ be a function defined and bounded on $[a, b]$. $f$ is said to be \textbf{integrable} if there exists one and only one number @@ -75,18 +75,18 @@ If $f$ is integrable on $[a, b]$, we say that the integral $\int_a^b f(x) \mathop{dx}$ \textbf{exists}. \section{\pending{Integral of a Bounded Function}}% -\label{ref:integral-bounded-function} +\hyperlabel{ref:integral-bounded-function} Let $f$ be a function defined and bounded on $[a, b]$. Let $s$ and $t$ denote arbitrary step functions defined on $[a, b]$ such that \begin{equation} - \label{ref:integral-bounded-function-eq1} + \hyperlabel{ref:integral-bounded-function-eq1} s(x) \leq f(x) \leq t(x) \end{equation} for every $x$ in $[a, b]$. If there is one and only one number $I$ such that \begin{equation} - \label{ref:integral-bounded-function-eq2} + \hyperlabel{ref:integral-bounded-function-eq2} \int_a^b s(x) \mathop{dx} \leq I \leq \int_a^b t(x) \mathop{dx} \end{equation} for every pair of step functions $s$ and $t$ satisfying @@ -103,7 +103,7 @@ The function $f$ is called the \textbf{integrand}, the numbers $a$ and $b$ are \textbf{interval of integration}. \section{\pending{Integral of a Step Function}}% -\label{ref:integral-step-function} +\hyperlabel{ref:integral-step-function} Let $s$ be a \nameref{ref:step-function} defined on $[a, b]$, and let $P = \{x_0, x_1, \ldots, x_n\}$ be a \nameref{ref:partition} of $[a, b]$ @@ -119,7 +119,7 @@ If $a < b$, we define $\int_b^a s(x) \mathop{dx} = -\int_a^b s(x) \mathop{dx}$. We also define $\int_a^a s(x) \mathop{dx} = 0$. \section{\pending{Lower Integral}}% -\label{ref:lower-integral} +\hyperlabel{ref:lower-integral} Let $f$ be a function bounded on $[a, b]$ and $S$ denote the set of numbers $\int_a^b s(x) \mathop{dx}$ obtained as $s$ runs through all @@ -129,7 +129,7 @@ The number $\sup{S}$ is called the \textbf{lower integral of $f$}. It is denoted as $\ubar{I}(f)$. \section{\pending{Monotonic}}% -\label{ref:monotonic} +\hyperlabel{ref:monotonic} A function $f$ is called \textbf{monotonic} on set $S$ if it is increasing on $S$ or if it is decreasing on $S$. @@ -143,13 +143,13 @@ In other words, $f$ is piecewise monotonic on $[a, b]$ if there is a the open subintervals of $P$. \section{\defined{Partition}}% -\label{ref:partition} +\hyperlabel{ref:partition} Let $[a, b]$ be a closed interval decomposed into $n$ subintervals by inserting $n - 1$ points of subdivision, say $x_1$, $x_2$, $\ldots$, $x_{n-1}$, subject only to the restriction \begin{equation} - \label{sec:partition-eq1} + \hyperlabel{sec:partition-eq1} a < x_1 < x_2 < \cdots < x_{n-1} < b. \end{equation} It is convenient to denote the point $a$ itself by $x_0$ and the point $b$ by @@ -165,7 +165,7 @@ A collection of points satisfying \eqref{sec:partition-eq1} is called a \end{definition} \section{\pending{Refinement}}% -\label{ref:refinement} +\hyperlabel{ref:refinement} Let $P$ be a \nameref{ref:partition} of closed interval $[a, b]$. A \textbf{refinement} $P'$ of $P$ is a partition formed by adjoining more @@ -175,7 +175,7 @@ $P'$ is said to be \textbf{finer than} $P$. The union of two partitions $P_1$ and $P_2$ is called the \textbf{common refinement} of $P_1$ and $P_2$. \section{\defined{Step Function}}% -\label{ref:step-function} +\hyperlabel{ref:step-function} A function $s$, whose domain is a closed interval $[a, b]$, is called a \textbf{step function} if there is a \nameref{ref:partition} @@ -195,7 +195,7 @@ That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$ \end{definition} \section{\defined{Supremum}}% -\label{ref:supremum} +\hyperlabel{ref:supremum} A number $B$ is called a \textbf{supremum} of a nonempty set $S$ if $B$ has the following two properties: @@ -212,7 +212,7 @@ Such a number $B$ is also known as the \textbf{least upper bound}. \end{definition} \section{\pending{Upper Integral}}% -\label{ref:upper-integral} +\hyperlabel{ref:upper-integral} Let $f$ be a function bounded on $[a, b]$ and $T$ denote the set of numbers $\int_a^b t(x) \mathop{dx}$ obtained as $t$ runs through all @@ -224,10 +224,10 @@ It is denoted as $\bar{I}(f)$. \endgroup \chapter{A Set of Axioms for the Real-Number System}% -\label{chap:set-axioms-real-number-system} +\hyperlabel{chap:set-axioms-real-number-system} \section{\verified{Lemma 1}}% -\label{sec:lemma-1} +\hyperlabel{sec:lemma-1} \begin{lemma}[1] @@ -250,8 +250,8 @@ It is denoted as $\bar{I}(f)$. \end{proof} \section{\verified{Existence of a Greatest Lower Bound}} -\label{sec:existence-greatest-lower-bound} -\label{sec:theorem-i.27} +\hyperlabel{sec:existence-greatest-lower-bound} +\hyperlabel{sec:theorem-i.27} \begin{theorem}[I.27] @@ -275,8 +275,8 @@ It is denoted as $\bar{I}(f)$. \end{proof} \section{\verified{Positive Integers Unbounded Above}}% -\label{sec:positive-integers-unbounded-above} -\label{sec:theorem-i.29} +\hyperlabel{sec:positive-integers-unbounded-above} +\hyperlabel{sec:theorem-i.29} \begin{theorem}[I.29] @@ -299,8 +299,8 @@ It is denoted as $\bar{I}(f)$. \end{proof} \section{\verified{Archimedean Property of the Reals}}% -\label{sec:archimedean-property-reals} -\label{sec:theorem-i.30} +\hyperlabel{sec:archimedean-property-reals} +\hyperlabel{sec:theorem-i.30} \begin{theorem}[I.30] @@ -322,7 +322,7 @@ It is denoted as $\bar{I}(f)$. \end{proof} \section{\verified{Theorem I.31}}% -\label{sec:theorem-i.31} +\hyperlabel{sec:theorem-i.31} \begin{theorem}[I.31] @@ -369,7 +369,7 @@ It is denoted as $\bar{I}(f)$. \end{proof} \section{\verified{Lemma 2}}% -\label{sec:lemma-2} +\hyperlabel{sec:lemma-2} \begin{lemma}[2] @@ -415,12 +415,12 @@ It is denoted as $\bar{I}(f)$. \end{proof} \section{\verified{Theorem I.32}}% -\label{sec:theorem-i.32} +\hyperlabel{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}}% -\label{sub:theorem-i.32a} +\hyperlabel{sub:theorem-i.32a} \begin{theorem}[I.32a] @@ -446,7 +446,7 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers. \end{proof} \subsection{\verified{Theorem I.32b}}% -\label{sub:theorem-i.32b} +\hyperlabel{sub:theorem-i.32b} \begin{theorem}[I.32b] @@ -472,8 +472,8 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers. \end{proof} \section{\verified{Additive Property of Supremums and Infimums}}% -\label{sec:additive-property-supremums-infimums} -\label{sec:theorem-i.33} +\hyperlabel{sec:additive-property-supremums-infimums} +\hyperlabel{sec:theorem-i.33} Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set $$C = \{a + b : a \in A, b \in B\}.$$ @@ -481,7 +481,7 @@ 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}}% -\label{sub:theorem-i.33a} +\hyperlabel{sub:theorem-i.33a} \begin{theorem}[I.33a] @@ -499,7 +499,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set $\sup{A} + \sup{B}$ is the \textit{least} upper bound of $C$. \paragraph{(i)}% - \label{par:theorem-i.33a-i} + \hyperlabel{par:theorem-i.33a-i} Let $x \in C$. By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such @@ -518,7 +518,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set Let $n > 0$ be an integer. We now prove that \begin{equation} - \label{par:theorem-i.33a-ii-eq1} + \hyperlabel{par:theorem-i.33a-ii-eq1} \sup{C} \leq \sup{A} + \sup{B} \leq \sup{C} + 1 / n. \end{equation} @@ -552,7 +552,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set \end{proof} \subsection{\verified{Theorem I.33b}}% -\label{sub:theorem-i.33b} +\hyperlabel{sub:theorem-i.33b} \begin{theorem}[I.33b] @@ -570,7 +570,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set $\inf{A} + \inf{B}$ is the \textit{greatest} lower bound of $C$. \paragraph{(i)}% - \label{par:theorem-i.33b-i} + \hyperlabel{par:theorem-i.33b-i} Let $x \in C$. By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such @@ -589,7 +589,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set Let $n > 0$ be an integer. We now prove that \begin{equation} - \label{par:theorem-i.33b-ii-eq1} + \hyperlabel{par:theorem-i.33b-ii-eq1} \inf{C} - 1 / n \leq \inf{A} + \inf{B} \leq \inf{C}. \end{equation} @@ -623,7 +623,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set \end{proof} \section{\verified{Theorem I.34}}% -\label{sec:theorem-i.34} +\hyperlabel{sec:theorem-i.34} \begin{theorem}[I.34] @@ -658,17 +658,17 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set \end{proof} \chapter{The Concepts of Integral Calculus}% -\label{chap:concepts-integral-calculus} +\hyperlabel{chap:concepts-integral-calculus} \section{The Concept of Area as a Set Function}% -\label{sec:concept-area-set-function} +\hyperlabel{sec:concept-area-set-function} We assume there exists a class $\mathscr{M}$ of measurable sets in the plane and a set function $a$, whose domain is $\mathscr{M}$, with the following properties: \subsection{\defined{Nonnegative Property}}% -\label{sub:nonnegative-property} +\hyperlabel{sub:nonnegative-property} For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$. @@ -680,7 +680,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$. \end{axiom} \subsection{\defined{Additive Property}}% -\label{sub:area-additive-property} +\hyperlabel{sub:area-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 $a(S \cup T) = a(S) + a(T) - a(S \cap T)$. @@ -693,7 +693,7 @@ If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in \end{axiom} \subsection{\defined{Difference Property}}% -\label{sub:area-difference-property} +\hyperlabel{sub:area-difference-property} If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in $\mathscr{M}$, and we have $a(T - S) = a(T) - a(S)$. @@ -706,7 +706,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in \end{axiom} \subsection{\defined{Invariance Under Congruence}}% -\label{sub:area-invariance-under-congruence} +\hyperlabel{sub:area-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 $a(S) = a(T)$. @@ -719,7 +719,7 @@ If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is \end{axiom} \subsection{\defined{Choice of Scale}}% -\label{sub:area-choice-scale} +\hyperlabel{sub:area-choice-scale} Every rectangle $R$ is in $\mathscr{M}$. If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$. @@ -732,12 +732,12 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$. \end{axiom} \subsection{\pending{Exhaustion Property}}% -\label{sub:area-exhaustion-property} +\hyperlabel{sub:area-exhaustion-property} Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so that \begin{equation} - \label{sub:exhaustion-property-eq1} + \hyperlabel{sub:exhaustion-property-eq1} S \subseteq Q \subseteq T. \end{equation} If there is one and only one number $c$ which satisfies the inequalities @@ -752,15 +752,15 @@ If there is one and only one number $c$ which satisfies the inequalities \end{axiom} \section{Exercises 1.7}% -\label{sec:exercises-1.7} +\hyperlabel{sec:exercises-1.7} \subsection{\pending{Exercise 1.7.1}}% -\label{sub:exercise-1.7.1} +\hyperlabel{sub:exercise-1.7.1} Prove that each of the following sets is measurable and has zero area: \subsubsection{\pending{Exercise 1.7.1a}}% -\label{ssub:exercise-1.7.1a} +\hyperlabel{ssub:exercise-1.7.1a} A set consisting of a single point. @@ -776,7 +776,7 @@ A set consisting of a single point. \end{proof} \subsubsection{\pending{Exercise 1.7.1b}}% -\label{ssub:exercise-1.7.1b} +\hyperlabel{ssub:exercise-1.7.1b} A set consisting of a finite number of points in a plane. @@ -808,7 +808,7 @@ A set consisting of a finite number of points in a plane. a(S_{k+1}) & = a(S_k \cup T) \nonumber \\ & = a(S_k) + a(T) - a(S_k \cap T) \nonumber \\ - & = 0 + 0 - a(S_k \cap T). \label{ssub:exercise-1.7.1b-eq1} + & = 0 + 0 - a(S_k \cap T). \hyperlabel{ssub:exercise-1.7.1b-eq1} \end{align} There are two cases to consider: @@ -835,7 +835,7 @@ A set consisting of a finite number of points in a plane. \end{proof} \subsubsection{\pending{Exercise 1.7.1c}}% -\label{ssub:exercise-1.7.1c} +\hyperlabel{ssub:exercise-1.7.1c} The union of a finite collection of line segments in a plane. @@ -871,7 +871,7 @@ The union of a finite collection of line segments in a plane. a(S_{k+1}) & = a(S_k \cup T) \nonumber \\ & = a(S_k) + a(T) - a(S_k \cap T) \nonumber \\ - & = 0 + 0 - a(S_k \cap T). \label{ssub:exercise-1.7.1c-eq1} + & = 0 + 0 - a(S_k \cap T). \hyperlabel{ssub:exercise-1.7.1c-eq1} \end{align} There are two cases to consider: @@ -898,7 +898,7 @@ The union of a finite collection of line segments in a plane. \end{proof} \subsection{\pending{Exercise 1.7.2}}% -\label{sub:exercise-1.7.2} +\hyperlabel{sub:exercise-1.7.2} Every right triangular region is measurable because it can be obtained as the intersection of two rectangles. @@ -949,7 +949,7 @@ Prove that every triangular region is measurable and that its area is one half \end{proof} \subsection{\pending{Exercise 1.7.3}}% -\label{sub:exercise-1.7.3} +\hyperlabel{sub:exercise-1.7.3} Prove that every trapezoid and every parallelogram is measurable and derive the usual formulas for their areas. @@ -1012,7 +1012,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the Then $T' \cup R$ is an acute trapezoid. By \nameref{sub:area-invariance-under-congruence}, \begin{equation} - \label{sub:exercise-1.7.3-eq1} + \hyperlabel{sub:exercise-1.7.3-eq1} \tag{3.1} a(T' \cup R) = a(T \cup R). \end{equation} @@ -1039,7 +1039,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the Then $T' \cup R$ is an acute trapezoid. By \nameref{sub:area-invariance-under-congruence}, \begin{equation} - \label{sub:exercise-1.7.3-eq2} + \hyperlabel{sub:exercise-1.7.3-eq2} a(T' \cup R) = a(T \cup R). \end{equation} By construction, $T' \cup R$ has height $h$ and bases $b - c$ and $b + c$ @@ -1054,14 +1054,14 @@ Prove that every trapezoid and every parallelogram is measurable and derive the \end{proof} \subsection{\pending{Exercise 1.7.4}}% -\label{sub:exercise-1.7.4} +\hyperlabel{sub:exercise-1.7.4} 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. \subsubsection{\pending{Exercise 1.7.4a}}% -\label{ssub:exercise-1.7.4a} +\hyperlabel{ssub:exercise-1.7.4a} Prove that the formula is valid for rectangles with sides parallel to the coordinate axes. @@ -1089,7 +1089,7 @@ Prove that the formula is valid for rectangles with sides parallel to the \end{proof} \subsubsection{\pending{Exercise 1.7.4b}}% -\label{ssub:exercise-1.7.4b} +\hyperlabel{ssub:exercise-1.7.4b} Prove that the formula is valid for right triangles and parallelograms. @@ -1111,12 +1111,12 @@ Prove that the formula is valid for right triangles and parallelograms. By construction, $T$ shares two sides with $R$. Therefore \begin{equation} - \label{ssub:exercise-1.7.4b-eq1} + \hyperlabel{ssub:exercise-1.7.4b-eq1} B_T = \frac{1}{2}B_R - 1 + H_L. \end{equation} Likewise, \begin{equation} - \label{ssub:exercise-1.7.4b-eq2} + \hyperlabel{ssub:exercise-1.7.4b-eq2} I_T = \frac{1}{2}(I_R - (H_L - 2)). \end{equation} The following shows the lattice point area formula is in agreement with @@ -1138,7 +1138,7 @@ Prove that the formula is valid for right triangles and parallelograms. \end{proof} \subsubsection{\pending{Exercise 1.7.4c}}% -\label{ssub:exercise-1.7.4c} +\hyperlabel{ssub:exercise-1.7.4c} Use induction on the number of edges to construct a proof for general polygons. @@ -1204,7 +1204,7 @@ Use induction on the number of edges to construct a proof for general polygons. \end{proof} \subsection{\pending{Exercise 1.7.5}}% -\label{sub:exercise-1.7.5} +\hyperlabel{sub:exercise-1.7.5} Prove that a triangle whose vertices are lattice points cannot be equilateral. @@ -1219,7 +1219,7 @@ ways, using Exercises 2 and 4.] Then $T$ has height $h = (a\sqrt{3}) / 2$. By \nameref{sub:exercise-1.7.2}, \begin{equation} - \label{sub:exercise-1.7.5-eq1} + \hyperlabel{sub:exercise-1.7.5-eq1} \tag{5.1} a(T) = \frac{1}{2}ah = \frac{a^2\sqrt{3}}{4}. \end{equation} @@ -1227,7 +1227,7 @@ ways, using Exercises 2 and 4.] $T$ respectively. By \nameref{sub:exercise-1.7.4}, \begin{equation} - \label{sub:exercise-1.7.5-eq2} + \hyperlabel{sub:exercise-1.7.5-eq2} \tag{5.2} a(T) = I + \frac{1}{2}B - 1. \end{equation} @@ -1240,7 +1240,7 @@ ways, using Exercises 2 and 4.] \end{proof} \subsection{\pending{Exercise 1.7.6}}% -\label{sub:exercise-1.7.6} +\hyperlabel{sub:exercise-1.7.6} Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all subsets of $A$. @@ -1322,15 +1322,15 @@ Prove that the set function $n$ satisfies the first three axioms for area. \end{proof} \section{Exercises 1.11}% -\label{sec:exercises-1-11} +\hyperlabel{sec:exercises-1-11} \subsection{\pending{Exercise 1.11.4}}% -\label{sub:exercise-1.11.4} +\hyperlabel{sub:exercise-1.11.4} Prove that the greatest-integer function has the properties indicated: \subsubsection{\verified{Exercise 1.11.4a}}% -\label{ssub:exercise-1.11.4a} +\hyperlabel{ssub:exercise-1.11.4a} $\floor{x + n} = \floor{x} + n$ for every integer $n$. @@ -1350,7 +1350,7 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$. \end{proof} \subsubsection{\verified{Exercise 1.11.4b}}% -\label{ssub:exercise-1.11.4b} +\hyperlabel{ssub:exercise-1.11.4b} $\floor{-x} = \begin{cases} @@ -1399,7 +1399,7 @@ $\floor{-x} = \end{proof} \subsubsection{\verified{Exercise 1.11.4c}}% -\label{ssub:exercise-1.11.4c} +\hyperlabel{ssub:exercise-1.11.4c} $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$. @@ -1416,7 +1416,7 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$. & = \floor{\floor{x} + \{x\} + \floor{y} + \{y\}} \nonumber \\ & = \floor{\floor{x} + \floor{y} + \{x\} + \{y\}} \nonumber \\ & = \floor{x} + \floor{y} + \floor{\{x\} + \{y\}} - & \textref{ssub:exercise-1.11.4a} \label{ssub:exercise-1.11.4c-eq1} + & \textref{ssub:exercise-1.11.4a} \hyperlabel{ssub:exercise-1.11.4c-eq1} \end{align} There are two cases to consider: @@ -1443,7 +1443,7 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$. \end{proof} \subsubsection{\pending{Exercise 1.11.4d}}% -\label{ssub:exercise-1.11.4d} +\hyperlabel{ssub:exercise-1.11.4d} $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$ @@ -1457,7 +1457,7 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$ \end{proof} \subsubsection{\pending{Exercise 1.11.4e}}% -\label{ssub:exercise-1.11.4e} +\hyperlabel{ssub:exercise-1.11.4e} $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$ @@ -1471,8 +1471,8 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$ \end{proof} \subsection{\pending{Hermite's Identity}}% -\label{sub:hermites-identity} -\label{sub:exercise-1.11.5} +\hyperlabel{sub:hermites-identity} +\hyperlabel{sub:exercise-1.11.5} The formulas in Exercises 4(d) and 4(e) suggest a generalization for $\floor{nx}$. @@ -1486,7 +1486,7 @@ State and prove such a generalization. We prove that for all natural numbers $n$ and real numbers $x$, the following identity holds: \begin{equation} - \label{sub:exercise-1.11.5-eq1} + \hyperlabel{sub:exercise-1.11.5-eq1} \floor{nx} = \sum_{i=0}^{n-1} \floor{x + \frac{i}{n}} \end{equation} By definition of the floor function, $x = \floor{x} + r$ for some @@ -1497,7 +1497,7 @@ State and prove such a generalization. By construction, $\cup\; S = \ico{0}{1}$. Therefore there exists some $j \in \mathbb{N}$ such that \begin{equation} - \label{sub:exercise-1.11.5-eq2} + \hyperlabel{sub:exercise-1.11.5-eq2} r \in \ico{\frac{j}{n}}{\frac{j+1}{n}}. \end{equation} With these definitions established, we now show the left- and right-hand sides @@ -1516,7 +1516,7 @@ State and prove such a generalization. & = \floor{n\floor{x}} + \floor{nr}. \nonumber & \textref{ssub:exercise-1.11.4a} \\ & = \floor{n\floor{x}} + j \nonumber \\ - & = n\floor{x} + j. \label{sub:exercise-1.11.5-eq3} + & = n\floor{x} + j. \hyperlabel{sub:exercise-1.11.5-eq3} \end{align} \paragraph{Right-Hand Side}% @@ -1530,7 +1530,7 @@ State and prove such a generalization. Letting $z$ denote the number of summands that contribute $\floor{x} + 1$, we have \begin{equation} - \label{sub:exercise-1.11.5-eq4} + \hyperlabel{sub:exercise-1.11.5-eq4} \sum_{i=0}^{n-1} \floor{x + \frac{i}{n}} = n\floor{x} + z. \end{equation} The value of $z$ corresponds to the number of indices $i$ that satisfy @@ -1547,7 +1547,7 @@ State and prove such a generalization. $z = (n - 1) - (n - j) + 1 = j$ of the $n$ summands. Substituting the value of $z$ into \eqref{sub:exercise-1.11.5-eq4} yields \begin{equation} - \label{sub:exercise-1.11.5-eq5} + \hyperlabel{sub:exercise-1.11.5-eq5} \sum_{i=0}^{n-1} \floor{x + \frac{i}{n}} = n\floor{x} + j. \end{equation} @@ -1560,7 +1560,7 @@ State and prove such a generalization. \end{proof} \subsection{\pending{Exercise 1.11.6}}% -\label{sub:exercise-1.11.6} +\hyperlabel{sub:exercise-1.11.6} Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are integers. @@ -1576,7 +1576,7 @@ Prove that the number of lattice points in $S$ is equal to the sum Let $i = a, \ldots, b$ and define $S_i = \mathbb{N} \cap \ioc{0}{f(i)}$. By construction, the number of lattice points in $S$ is \begin{equation} - \label{sub:exercise-1.11.6-eq1} + \hyperlabel{sub:exercise-1.11.6-eq1} \sum_{n = a}^b \abs{S_n}. \end{equation} All that remains is to show $\abs{S_i} = \floor{f(i)}$. @@ -1603,7 +1603,7 @@ Prove that the number of lattice points in $S$ is equal to the sum \end{proof} \subsection{\pending{Exercise 1.11.7}}% -\label{sub:exercise-1.11.7} +\hyperlabel{sub:exercise-1.11.7} If $a$ and $b$ are positive integers with no common factor, we have the formula $$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$ @@ -1613,7 +1613,7 @@ When $b = 1$, the sum on the left is understood to be $0$. assumption $b > 1$.} \subsubsection{\pending{Exercise 1.11.7a}}% -\label{ssub:exercise-1.11.7a} +\hyperlabel{ssub:exercise-1.11.7a} Derive this result by a geometric argument, counting lattice points in a right triangle. @@ -1626,7 +1626,7 @@ Derive this result by a geometric argument, counting lattice points in a right By \nameref{sub:exercise-1.11.6}, the number of lattice points of $S$ is equal to the sum \begin{equation} - \label{ssub:exercise-1.11.7a-eq1} + \hyperlabel{ssub:exercise-1.11.7a-eq1} \sum_{n=1}^{b-1} \floor{f(n)} = \sum_{n=1}^{b-1} \floor{\frac{na}{b}}. \end{equation} Define $T$ to be the triangle of width $w = b$ and height $h = f(b) = a$ @@ -1638,7 +1638,7 @@ Derive this result by a geometric argument, counting lattice points in a right lattice points. \paragraph{(i)}% - \label{par:exercise-1.11.7a-i} + \hyperlabel{par:exercise-1.11.7a-i} Consider the line $L$ overlapping the hypotenuse of $T$. By construction, $T$'s hypotenuse has endpoints $(0, 0)$ and $(b, a)$. @@ -1665,13 +1665,13 @@ Derive this result by a geometric argument, counting lattice points in a right & = \frac{1}{2}(I_R - (H_L - 2)) \nonumber \\ & = \frac{1}{2}(I_R - (2 - 2)) & \textref{par:exercise-1.11.7a-i} \nonumber \\ - & = \frac{1}{2}I_R. & \label{ssub:exercise-1.11.7a-eq2} + & = \frac{1}{2}I_R. & \hyperlabel{ssub:exercise-1.11.7a-eq2} \end{align} Furthermore, since both the adjacent and opposite side of $T$ are not included in $T$ and there exist no lattice points on $T$'s hypotenuse besides the endpoints, it follows \begin{equation} - \label{ssub:exercise-1.11.7a-eq3} + \hyperlabel{ssub:exercise-1.11.7a-eq3} B_T = 0. \end{equation} Thus the number of lattice points of $T$ equals @@ -1680,7 +1680,7 @@ Derive this result by a geometric argument, counting lattice points in a right & = I_T & \eqref{ssub:exercise-1.11.7a-eq3} \nonumber \\ & = \frac{1}{2}I_R & \eqref{ssub:exercise-1.11.7a-eq2} \nonumber \\ & = \frac{(b - 1)(a - 1)}{2}. - & \textref{ssub:exercise-1.7.4a} \label{ssub:exercise-1.11.7a-eq4} + & \textref{ssub:exercise-1.7.4a} \hyperlabel{ssub:exercise-1.11.7a-eq4} \end{align} \paragraph{Conclusion}% @@ -1695,7 +1695,7 @@ Derive this result by a geometric argument, counting lattice points in a right \end{proof} \subsubsection{\pending{Exercise 1.11.7b}}% -\label{ssub:exercise-1.11.7b} +\hyperlabel{ssub:exercise-1.11.7b} Derive the result analytically as follows: By changing the index of summation, note that @@ -1713,7 +1713,7 @@ Now apply Exercises 4(a) and (b) to the bracket on the right. Thus $an / b$ is not an integer. By \nameref{ssub:exercise-1.11.4b}, \begin{equation} - \label{ssub:exercise-1.11.7b-eq1} + \hyperlabel{ssub:exercise-1.11.7b-eq1} \floor{-\frac{an}{b}} = -\floor{\frac{an}{b}} - 1. \end{equation} Consider the following: @@ -1737,7 +1737,7 @@ Now apply Exercises 4(a) and (b) to the bracket on the right. \end{proof} \subsection{\pending{Exercise 1.11.8}}% -\label{sub:exercise-1.11.8} +\hyperlabel{sub:exercise-1.11.8} Let $S$ be a set of points on the real line. Let $\mathcal{X}_S$ denote the \nameref{ref:characteristic-function} of $S$. @@ -1772,11 +1772,11 @@ This property is described by saying that every step function is a linear \end{proof} \section{Properties of the Integral of a Step Function}% -\label{sec:properties-integral-step-function} +\hyperlabel{sec:properties-integral-step-function} \subsection{\pending{Additive Property}}% -\label{sub:step-additive-property} -\label{sub:theorem-1.2} +\hyperlabel{sub:step-additive-property} +\hyperlabel{sub:theorem-1.2} \begin{theorem}[1.2] @@ -1818,8 +1818,8 @@ This property is described by saying that every step function is a linear \end{proof} \subsection{\pending{Homogeneous Property}}% -\label{sub:step-homogeneous-property} -\label{sub:theorem-1.3} +\hyperlabel{sub:step-homogeneous-property} +\hyperlabel{sub:theorem-1.3} \begin{theorem}[1.3] @@ -1849,8 +1849,8 @@ This property is described by saying that every step function is a linear \end{proof} \subsection{\pending{Linearity Property}}% -\label{sub:step-linearity-property} -\label{sub:theorem-1.4} +\hyperlabel{sub:step-linearity-property} +\hyperlabel{sub:theorem-1.4} \begin{theorem}[1.4] @@ -1879,8 +1879,8 @@ This property is described by saying that every step function is a linear \end{proof} \subsection{\pending{Comparison Theorem}}% -\label{sub:step-comparison-theorem} -\label{sub:theorem-1.5} +\hyperlabel{sub:step-comparison-theorem} +\hyperlabel{sub:theorem-1.5} \begin{theorem}[1.5] @@ -1918,8 +1918,8 @@ This property is described by saying that every step function is a linear \end{proof} \subsection{\pending{Additivity With Respect to the Interval of Integration}}% -\label{sub:step-additivity-with-respect-interval-integration} -\label{sub:theorem-1.6} +\hyperlabel{sub:step-additivity-with-respect-interval-integration} +\hyperlabel{sub:theorem-1.6} \begin{theorem}[1.6] @@ -1964,8 +1964,8 @@ This property is described by saying that every step function is a linear \end{proof} \subsection{\pending{Invariance Under Translation}}% -\label{sub:step-invariance-under-translation} -\label{sub:theorem-1.7} +\hyperlabel{sub:step-invariance-under-translation} +\hyperlabel{sub:theorem-1.7} \begin{theorem}[1.7] @@ -2005,8 +2005,8 @@ This property is described by saying that every step function is a linear \end{proof} \subsection{\pending{Expansion or Contraction of the Interval of Integration}}% -\label{sub:step-expansion-contraction-interval-integration} -\label{sub:theorem-1.8} +\hyperlabel{sub:step-expansion-contraction-interval-integration} +\hyperlabel{sub:theorem-1.8} \begin{theorem}[1.8] @@ -2065,7 +2065,7 @@ This property is described by saying that every step function is a linear \end{proof} \subsection{\pending{Reflection Property}}% -\label{sub:step-reflection-property} +\hyperlabel{sub:step-reflection-property} Let $s$ be a step function on closed interval $[a, b]$. Then @@ -2083,15 +2083,15 @@ Then \end{proof} \section{Exercises 1.15}% -\label{sec:exercises-1.15} +\hyperlabel{sec:exercises-1.15} \subsection{\pending{Exercise 1.15.1}}% -\label{sub:exercise-1.15.1} +\hyperlabel{sub:exercise-1.15.1} Compute the value of each of the following integrals. \subsubsection{\pending{Exercise 1.15.1a}}% -\label{ssub:exercise-1.15.1a} +\hyperlabel{ssub:exercise-1.15.1a} $\int_{-1}^3 \floor{x} \mathop{dx}$. @@ -2113,7 +2113,7 @@ $\int_{-1}^3 \floor{x} \mathop{dx}$. \end{proof} \subsubsection{\pending{Exercise 1.15.1c}}% -\label{ssub:exercise-1.15.1c} +\hyperlabel{ssub:exercise-1.15.1c} $\int_{-1}^3 \left(\floor{x} + \floor{x + \frac{1}{2}}\right) \mathop{dx}$. @@ -2141,7 +2141,7 @@ $\int_{-1}^3 \left(\floor{x} + \floor{x + \frac{1}{2}}\right) \mathop{dx}$. \end{proof} \subsubsection{\pending{Exericse 1.15.1e}}% -\label{ssub:exercise-1.15.1e} +\hyperlabel{ssub:exercise-1.15.1e} $\int_{-1}^3 \floor{2x} \mathop{dx}$. @@ -2156,7 +2156,7 @@ $\int_{-1}^3 \floor{2x} \mathop{dx}$. \end{proof} \subsection{\pending{Exercise 1.15.3}}% -\label{sub:exercise-1.15.3} +\hyperlabel{sub:exercise-1.15.3} Show that $\int_a^b \floor{x} \mathop{dx} + \int_a^b \floor{-x} \mathop{dx} = a - b$. @@ -2187,10 +2187,10 @@ Show that \end{proof} \subsection{\pending{Exercise 1.15.5}}% -\label{sub:exercise-1.15.5} +\hyperlabel{sub:exercise-1.15.5} \subsubsection{\pending{Exercise 1.15.5a}}% -\label{ssub:exercise-1.15.5a} +\hyperlabel{ssub:exercise-1.15.5a} Prove that $\int_0^2 \floor{t^2} \mathop{dt} = 5 - \sqrt{2} - \sqrt{3}$. @@ -2213,7 +2213,7 @@ Prove that $\int_0^2 \floor{t^2} \mathop{dt} = 5 - \sqrt{2} - \sqrt{3}$. \end{proof} \subsubsection{\pending{Exercise 1.15.5b}}% -\label{ssub:exercise-1.15.5b} +\hyperlabel{ssub:exercise-1.15.5b} Compute $\int_{-3}^3 \floor{t^2} \mathop{dt}$. @@ -2235,12 +2235,12 @@ Compute $\int_{-3}^3 \floor{t^2} \mathop{dt}$. & = \sum_{k=1}^9 s_k \cdot (x_k - x_{k-1}) \nonumber \\ & = \sum_{k=0}^8 k \cdot (\sqrt{k + 1} - \sqrt{k}). - \label{sub:exercise-1.15.5b-eq1} + \hyperlabel{sub:exercise-1.15.5b-eq1} \end{align} We notice $\floor{t^2}$ is symmetric about the $y$-axis. Thus \begin{equation} - \label{sub:exercise-1.15.5b-eq2} + \hyperlabel{sub:exercise-1.15.5b-eq2} \int_{-3}^0 \floor{t^2} \mathop{dt} = \int_0^3 \floor{t^2}. \end{equation} By \nameref{sub:step-additivity-with-respect-interval-integration}, @@ -2257,10 +2257,10 @@ Compute $\int_{-3}^3 \floor{t^2} \mathop{dt}$. \end{proof} \subsection{\pending{Exercise 1.15.7}}% -\label{sub:exercise-1.15.7} +\hyperlabel{sub:exercise-1.15.7} \subsubsection{\pending{Exercise 1.15.7a}}% -\label{ssub:exercise-1.15.7a} +\hyperlabel{ssub:exercise-1.15.7a} Compute $\int_0^9 \floor{\sqrt{t}} \mathop{dt}$. @@ -2282,7 +2282,7 @@ Compute $\int_0^9 \floor{\sqrt{t}} \mathop{dt}$. \end{proof} \subsubsection{\pending{Exercise 1.15.7b}}% -\label{ssub:exercise-1.15.7b} +\hyperlabel{ssub:exercise-1.15.7b} If $n$ is a positive integer, prove that $$\int_0^{n^2} \floor{\sqrt{t}} \mathop{dt} = n(n - 1)(4n + 1) / 6.$$ @@ -2291,7 +2291,7 @@ If $n$ is a positive integer, prove that Define predicate $P(n)$ as \begin{equation} - \label{sub:exercise-1.15.7b-eq1} + \hyperlabel{sub:exercise-1.15.7b-eq1} \int_0^{n^2} \floor{\sqrt{t}} \mathop{dt} = \frac{n(n - 1)(4n + 1)}{6}. \end{equation} We use induction to prove $P(n)$ holds for all integers satisfying $n > 0$. @@ -2357,12 +2357,12 @@ If $n$ is a positive integer, prove that \end{proof} \subsection{\pending{Exercise 1.15.9}}% -\label{sub:exercise-1.15.9} +\hyperlabel{sub:exercise-1.15.9} Show that the following property is equivalent to \nameref{sub:step-expansion-contraction-interval-integration}: \begin{equation} - \label{sub:exercise-1.15.9-eq1} + \hyperlabel{sub:exercise-1.15.9-eq1} \int_{ka}^{kb} f(x) \mathop{dx} = k \int_a^b f(kx) \mathop{dx}. \end{equation} @@ -2379,18 +2379,18 @@ Show that the following property is equivalent to \end{proof} \subsection{\pending{Exercise 1.15.11}}% -\label{sub:exercise-1.15.11} +\hyperlabel{sub:exercise-1.15.11} If we instead defined the integral of step functions as \begin{equation*} - \label{sub:exercise-1.15.11-eq1} + \hyperlabel{sub:exercise-1.15.11-eq1} \int_a^b s(x) \mathop{dx} = \sum_{k=1}^n s_k^3 \cdot (x_k - x_{k-1}), \end{equation*} a new and different theory of integration would result. Which of the following properties would remain valid in this new theory? \subsubsection{\pending{Exercise 1.15.11a}}% -\label{ssub:exercise-1.15.11a} +\hyperlabel{ssub:exercise-1.15.11a} $\int_a^b s + \int_b^c s = \int_a^c s$. @@ -2426,7 +2426,7 @@ $\int_a^b s + \int_b^c s = \int_a^c s$. \end{proof} \subsubsection{\pending{Exercise 1.15.11b}}% -\label{ssub:exercise-1.15.11b} +\hyperlabel{ssub:exercise-1.15.11b} $\int_a^b (s + t) = \int_a^b s + \int_a^b t$. @@ -2473,7 +2473,7 @@ $\int_a^b (s + t) = \int_a^b s + \int_a^b t$. \end{proof} \subsubsection{\pending{Exercise 1.15.11c}}% -\label{ssub:exercise-1.15.11c} +\hyperlabel{ssub:exercise-1.15.11c} $\int_a^b c \cdot s = c \int_a^b s$. @@ -2505,7 +2505,7 @@ $\int_a^b c \cdot s = c \int_a^b s$. \end{proof} \subsubsection{\pending{Exercise 1.15.11d}}% -\label{ssub:exercise-1.15.11d} +\hyperlabel{ssub:exercise-1.15.11d} $\int_{a+c}^{b+c} s(x) \mathop{dx} = \int_a^b s(x + c) \mathop{dx}$. @@ -2543,7 +2543,7 @@ $\int_{a+c}^{b+c} s(x) \mathop{dx} = \int_a^b s(x + c) \mathop{dx}$. \end{proof} \subsubsection{\pending{Exercise 1.15.11e}}% -\label{ssub:exercise-1.15.11e} +\hyperlabel{ssub:exercise-1.15.11e} If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. @@ -2581,10 +2581,10 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \end{proof} \section{Upper and Lower Integrals}% -\label{sec:upper-lower-integrals} +\hyperlabel{sec:upper-lower-integrals} \subsection{\pending{Theorem 1.9}}% -\label{sub:theorem-1.9} +\hyperlabel{sub:theorem-1.9} \begin{theorem}[1.9] @@ -2592,7 +2592,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. $\ubar{I}(f)$ and an upper integral $\overline{I}(f)$ satisfying the inequalities \begin{equation} - \label{sub:theorem-1.9-eq1} + \hyperlabel{sub:theorem-1.9-eq1} \int_a^b s(x) \mathop{dx} \leq \ubar{I}(f) \leq \bar{I}(f) \leq \int_a^b t(x) \mathop{dx} \end{equation} @@ -2647,10 +2647,10 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \end{proof} \section{The Area of an Ordinate Set Expressed as an Integral}% -\label{sec:area-ordinate-set-expressed-integral} +\hyperlabel{sec:area-ordinate-set-expressed-integral} \subsection{\pending{Theorem 1.10}}% -\label{sub:theorem-1.10} +\hyperlabel{sub:theorem-1.10} \begin{theorem}[1.10] @@ -2677,14 +2677,14 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \end{proof} \subsection{\pending{Theorem 1.11}}% -\label{sub:theorem-1.11} +\hyperlabel{sub:theorem-1.11} \begin{theorem}[1.11] Let $f$ be a nonnegative function, integrable on an interval $[a, b]$. Then the graph of $f$, that is, the set \begin{equation} - \label{sub:theorem-1.11-eq1} + \hyperlabel{sub:theorem-1.11-eq1} \{(x, y) \mid a \leq x \leq b, y = f(x)\}, \end{equation} is measurable and has area equal to $0$. @@ -2700,7 +2700,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. equal to $0$. \paragraph{(i)}% - \label{par:theorem-1.11-i} + \hyperlabel{par:theorem-1.11-i} By definition of integrability, there exists one and only one number $I$ such that @@ -2731,10 +2731,10 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \section [Integrability of Bounded Monotonic Functions] {Integrability of Bounded Monotonic \texorpdfstring{\\}{}Functions} -\label{sec:integrability-bounded-monotonic-functions} +\hyperlabel{sec:integrability-bounded-monotonic-functions} \subsection{\pending{Theorem 1.12}}% -\label{sub:theorem-1.12} +\hyperlabel{sub:theorem-1.12} \begin{theorem}[1.12] @@ -2766,7 +2766,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. on every $k$th open subinterval of $P$. Then, by \eqref{sub:theorem-1.9-eq1}, it follows \begin{equation} - \label{sub:theorem-1.12-eq1} + \hyperlabel{sub:theorem-1.12-eq1} \int_a^b s(x) \mathop{dx} \leq \ubar{I}(f) \leq \bar{I}(f) \leq \int_a^b t(x) \mathop{dx}. \end{equation} @@ -2805,7 +2805,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. on every $k$th open subinterval of $P$. Then, by \eqref{sub:theorem-1.9-eq1}, it follows \begin{equation} - \label{sub:theorem-1.12-eq2} + \hyperlabel{sub:theorem-1.12-eq2} \int_a^b s(x) \mathop{dx} \leq \ubar{I}(f) \leq \bar{I}(f) \leq \int_a^b t(x) \mathop{dx}. \end{equation} @@ -2838,7 +2838,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \end{proof} \subsection{\pending{Theorem 1.13}}% -\label{sub:theorem-1.13} +\hyperlabel{sub:theorem-1.13} \begin{theorem}[1.13] @@ -2846,7 +2846,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. Let $x_k = a + k(b - a) / n$ for $k = 0, 1, \ldots, n$. If $I$ is any number which satisfies the inequalities \begin{equation} - \label{sub:theorem-1.13-eq1} + \hyperlabel{sub:theorem-1.13-eq1} \frac{b - a}{n} \sum_{k=0}^{n-1} f(x_k) \leq I \leq \frac{b - a}{n} \sum_{k=1}^n f(x_k) @@ -2873,13 +2873,13 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \end{align*} Therefore \eqref{sub:theorem-1.13-eq1} can alternatively be written as \begin{equation} - \label{sub:theorem-1.13-eq2} + \hyperlabel{sub:theorem-1.13-eq2} \int_a^b s(x) \mathop{dx} \leq I \leq \int_a^b t(x) \mathop{dx}. \end{equation} By \nameref{sub:theorem-1.12}, $f$ is integrable. Therefore \nameref{sub:theorem-1.9} indicates $f$ satisfies \begin{equation} - \label{sub:theorem-1.13-eq3} + \hyperlabel{sub:theorem-1.13-eq3} \int_a^b s(x) \mathop{dx} \leq \int_a^b f(x) \mathop{dx} \leq \int_a^b t(x) \mathop{dx}. @@ -2910,7 +2910,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \end{proof} \subsection{\pending{Theorem 1.14}}% -\label{sub:theorem-1.14} +\hyperlabel{sub:theorem-1.14} \begin{theorem}[1.14] @@ -2918,7 +2918,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. Let $x_k = a + k(b - a) / n$ for $k = 0, 1, \ldots, n$. If $I$ is any number which satisfies the inequalities \begin{equation} - \label{sub:theorem-1.14-eq1} + \hyperlabel{sub:theorem-1.14-eq1} \frac{b - a}{n} \sum_{k=1}^n f(x_k) \leq I \leq \frac{b - a}{n} \sum_{k=0}^{n-1} f(x_k) @@ -2945,13 +2945,13 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \end{align*} Therefore \eqref{sub:theorem-1.14-eq1} can alternatively be written as \begin{equation} - \label{sub:theorem-1.14-eq2} + \hyperlabel{sub:theorem-1.14-eq2} \int_a^b s(x) \mathop{dx} \leq I \leq \int_a^b t(x) \mathop{dx}. \end{equation} By \nameref{sub:theorem-1.12}, $f$ is integrable. Therefore \nameref{sub:theorem-1.9} indicates $f$ satisfies \begin{equation} - \label{sub:theorem-1.14-eq3} + \hyperlabel{sub:theorem-1.14-eq3} \int_a^b s(x) \mathop{dx} \leq \int_a^b f(x) \mathop{dx} \leq \int_a^b t(x) \mathop{dx}. @@ -2984,8 +2984,8 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \subsection{\sorry{% Integral of \texorpdfstring{$\int_0^b x^p \mathop{dx}$}{int-x-p} when \texorpdfstring{$p$}{p} is a Positive Integer}}% -\label{sub:calculation-integral-int-x-p-p-positive-integer} -\label{sub:theorem-1.15} +\hyperlabel{sub:calculation-integral-int-x-p-p-positive-integer} +\hyperlabel{sub:theorem-1.15} \begin{theorem}[1.15] diff --git a/Bookshelf/Enderton/Logic.tex b/Bookshelf/Enderton/Logic.tex index b79caf4..0262d80 100644 --- a/Bookshelf/Enderton/Logic.tex +++ b/Bookshelf/Enderton/Logic.tex @@ -13,7 +13,7 @@ \renewcommand\thechapter{R} \chapter{Reference}% -\label{chap:reference} +\hyperlabel{chap:reference} \endgroup @@ -21,10 +21,10 @@ \setcounter{chapter}{0} \addtocounter{chapter}{-1} \chapter{Useful Facts About Sets}% -\label{chap:useful-facts-about-sets} +\hyperlabel{chap:useful-facts-about-sets} \section{\sorry{Lemma 0A}}% -\label{sec:lemma-0a} +\hyperlabel{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/Bookshelf/Enderton/Set.tex b/Bookshelf/Enderton/Set.tex index 4d51995..6cbc510 100644 --- a/Bookshelf/Enderton/Set.tex +++ b/Bookshelf/Enderton/Set.tex @@ -20,10 +20,10 @@ \addtocounter{chapter}{-1} \chapter{Reference}% -\label{chap:reference} +\hyperlabel{chap:reference} \section{\defined{Axiom of Choice, First Form}}% -\label{ref:axiom-of-choice-1} +\hyperlabel{ref:axiom-of-choice-1} For any relation $R$ there is a function $H \subseteq R$ with $\dom{H} = \dom{R}$. @@ -35,7 +35,7 @@ For any relation $R$ there is a function $H \subseteq R$ with \end{axiom} \section{\defined{Axiom of Choice, Second Form}}% -\label{ref:axiom-of-choice-2} +\hyperlabel{ref:axiom-of-choice-2} For any set $I$ and any function $H$ with domain $I$, if $H(i) \neq \emptyset$ for all $i \in I$, then $$\bigtimes_{i \in I} H(i) \neq \emptyset.$$ @@ -47,7 +47,7 @@ For any set $I$ and any function $H$ with domain $I$, if $H(i) \neq \emptyset$ \end{axiom} \section{\defined{Cartesian Product}}% -\label{ref:cartesian-product} +\hyperlabel{ref:cartesian-product} Let $I$ be a set and let $H$ be a \nameref{ref:function} whose domain includes $I$. Then for each $i$ in $I$ we have the set $H(i)$. @@ -63,7 +63,7 @@ We define the \textbf{cartesian product} of the $H(i)$'s as \end{definition} \section{\defined{Compatible}}% -\label{ref:compatible} +\hyperlabel{ref:compatible} A \nameref{ref:function} $F$ is \textbf{compatible} with relation $R$ if and only if for all $x$ and $y$ in $A$, @@ -76,7 +76,7 @@ A \nameref{ref:function} $F$ is \textbf{compatible} with relation $R$ if and \end{definition} \section{\defined{Composition}}% -\label{ref:composition} +\hyperlabel{ref:composition} The \textbf{composition} of sets $F$ and $G$ is $$F \circ G = \{\pair{u, v} \mid \exists t(uGt \land tFv)\}.$$ @@ -88,7 +88,7 @@ The \textbf{composition} of sets $F$ and $G$ is \end{definition} \section{\defined{Domain}}% -\label{ref:domain} +\hyperlabel{ref:domain} The \textbf{domain} of set $R$, denoted $\dom{R}$, is given by $$x \in \dom{R} \iff \exists y \pair{x, y} \in R.$$ @@ -100,7 +100,7 @@ The \textbf{domain} of set $R$, denoted $\dom{R}$, is given by \end{definition} \section{\defined{Empty Set Axiom}}% -\label{ref:empty-set-axiom} +\hyperlabel{ref:empty-set-axiom} There is a set having no members: $$\exists B, \forall x, x \not\in B.$$ @@ -112,7 +112,7 @@ There is a set having no members: \end{axiom} \section{\defined{Equivalence Class}}% -\label{ref:equivalence-class} +\hyperlabel{ref:equivalence-class} The set $[x]_R$ is defined by $$[x]_R = \{t \mid xRt\}.$$ If $R$ is an \nameref{ref:equivalence-relation} and $x \in fld R$, then $[x]_R$ @@ -126,7 +126,7 @@ If the relation $R$ is fixed by the context, we may write just $[x]$. \end{definition} \section{\defined{Equivalence Relation}}% -\label{ref:equivalence-relation} +\hyperlabel{ref:equivalence-relation} Relation $R$ is an \textbf{equivalence relation} on set $A$ if and only if $R$ is a binary \nameref{ref:relation} on $A$ that is \nameref{ref:reflexive} @@ -139,7 +139,7 @@ Relation $R$ is an \textbf{equivalence relation} on set $A$ if and only if \end{definition} \section{\defined{Extensionality Axiom}}% -\label{ref:extensionality-axiom} +\hyperlabel{ref:extensionality-axiom} If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, @@ -152,7 +152,7 @@ If two sets have exactly the same members, then they are equal: \end{axiom} \section{\defined{Field}}% -\label{ref:field} +\hyperlabel{ref:field} Given \nameref{ref:relation} $R$, the \textbf{field} of $R$, denoted $\fld{R}$, is given by $$\fld{R} = \dom{R} \cup \ran{R}.$$ @@ -164,7 +164,7 @@ Given \nameref{ref:relation} $R$, the \textbf{field} of $R$, denoted $\fld{R}$, \end{definition} \section{\defined{Function}}% -\label{ref:function} +\hyperlabel{ref:function} A \textbf{function} is a relation $F$ such that for each $x$ in $\dom{F}$ there is only one $y$ such that $xFy$. @@ -192,7 +192,7 @@ One-to-one functions are sometimes called \textbf{injections}. \end{definition} \section{\defined{Image}}% -\label{ref:image} +\hyperlabel{ref:image} Let $A$ and $F$ be arbitrary sets. The \textbf{image of $A$ under $F$} is the set @@ -209,7 +209,7 @@ The \textbf{image of $A$ under $F$} is the set \end{definition} \section{\defined{Inverse}}% -\label{ref:inverse} +\hyperlabel{ref:inverse} The \textbf{inverse} of a set $F$ is the set $$F^{-1} = \{\pair{u, v} \mid vFu\}.$$ @@ -221,7 +221,7 @@ The \textbf{inverse} of a set $F$ is the set \end{definition} \section{\defined{Ordered Pair}}% -\label{ref:ordered-pair} +\hyperlabel{ref:ordered-pair} For any sets $u$ and $v$, the \textbf{ordered pair} $\pair{u, v}$ is the set $\{\{u\}, \{u, v\}\}$. @@ -233,7 +233,7 @@ For any sets $u$ and $v$, the \textbf{ordered pair} $\pair{u, v}$ is \end{definition} \section{\defined{Pair Set}}% -\label{ref:pair-set} +\hyperlabel{ref:pair-set} For any sets $u$ and $v$, the \textbf{pair set $\{u, v\}$} is the set whose only members are $u$ and $v$. @@ -249,7 +249,7 @@ For any sets $u$ and $v$, the \textbf{pair set $\{u, v\}$} is the set whose \end{definition} \section{\defined{Pairing Axiom}}% -\label{ref:pairing-axiom} +\hyperlabel{ref:pairing-axiom} For any sets $u$ and $v$, there is a set having as members just $u$ and $v$: $$\forall u, \forall v, \exists B, \forall x, @@ -266,7 +266,7 @@ For any sets $u$ and $v$, there is a set having as members just $u$ and $v$: \end{axiom} \section{\defined{Partition}}% -\label{ref:partition} +\hyperlabel{ref:partition} A \textbf{partition} $\Pi$ of a set $A$ is a set of nonempty subsets of $A$ that is disjoint and exhaustive, i.e. @@ -282,7 +282,7 @@ A \textbf{partition} $\Pi$ of a set $A$ is a set of nonempty subsets of $A$ that \end{definition} \section{\defined{Power Set}}% -\label{ref:power-set} +\hyperlabel{ref:power-set} For any set $a$, the \textbf{power set $\powerset{a}$} is the set whose members are exactly the subsets of $a$. @@ -294,7 +294,7 @@ For any set $a$, the \textbf{power set $\powerset{a}$} is the set whose members \end{definition} \section{\defined{Power Set Axiom}}% -\label{ref:power-set-axiom} +\hyperlabel{ref:power-set-axiom} For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \iff x \subseteq a).$$ @@ -306,7 +306,7 @@ For any set $a$, there is a set whose members are exactly the subsets of $a$: \end{axiom} \section{\defined{Quotient Set}}% -\label{ref:quotient-set} +\hyperlabel{ref:quotient-set} If $R$ is an \nameref{ref:equivalence-relation} on set $A$, then we can define the \textbf{quotient set} $$A / R = \{[x]_R \mid x \in A\}$$ whose members are @@ -320,7 +320,7 @@ The expression $A / R$ is read "$A$ modulo $R$. \end{definition} \section{\defined{Range}}% -\label{ref:range} +\hyperlabel{ref:range} The \textbf{range} of set $R$, denoted $\ran{R}$, is given by $$x \in \ran{R} \iff \exists t \pair{t, x} \in R.$$ @@ -332,7 +332,7 @@ The \textbf{range} of set $R$, denoted $\ran{R}$, is given by \end{definition} \section{\defined{Reflexive}}% -\label{ref:reflexive} +\hyperlabel{ref:reflexive} A binary relation $R$ is \textbf{reflexive} on $A$ if and only if $xRx$ for all $x \in A$. @@ -344,7 +344,7 @@ A binary relation $R$ is \textbf{reflexive} on $A$ if and only if $xRx$ for all \end{definition} \section{\defined{Relation}}% -\label{ref:relation} +\hyperlabel{ref:relation} A \textbf{relation} is a set of \nameref{ref:ordered-pair}s. @@ -355,7 +355,7 @@ A \textbf{relation} is a set of \nameref{ref:ordered-pair}s. \end{definition} \section{\defined{Restriction}}% -\label{ref:restriction} +\hyperlabel{ref:restriction} The \textbf{restriction} of a set $F$ to set $A$ is the set $$F \restriction A = \{\pair{u, v} \mid uFv \land u \in A\}.$$ @@ -367,7 +367,7 @@ The \textbf{restriction} of a set $F$ to set $A$ is the set \end{definition} \section{\defined{Subset Axioms}}% -\label{ref:subset-axioms} +\hyperlabel{ref:subset-axioms} For each formula $\phi$ not containing $B$, the following is an axiom: $$\forall t_1, \cdots \forall t_k, \forall c, @@ -380,7 +380,7 @@ For each formula $\phi$ not containing $B$, the following is an axiom: \end{axiom} \section{\defined{Symmetric}}% -\label{ref:symmetric} +\hyperlabel{ref:symmetric} A binary relation $R$ is \textbf{symmetric} if and only if whenever $xRy$ then $yRx$. @@ -392,7 +392,7 @@ A binary relation $R$ is \textbf{symmetric} if and only if whenever $xRy$ then \end{definition} \section{\defined{Symmetric Difference}}% -\label{ref:symmetric-difference} +\hyperlabel{ref:symmetric-difference} The \textbf{symmetric difference} $A + B$ of sets $A$ and $B$ is the set $(A - B) \cup (B - A)$. @@ -404,7 +404,7 @@ The \textbf{symmetric difference} $A + B$ of sets $A$ and $B$ is the set \end{definition} \section{\defined{Transitive}}% -\label{ref:transitive} +\hyperlabel{ref:transitive} A binary relation $R$ is \textbf{transitive} if and only if whenever $xRy$ and $yRz$, then $xRz$. @@ -416,7 +416,7 @@ A binary relation $R$ is \textbf{transitive} if and only if whenever $xRy$ and \end{definition} \section{\defined{Union Axiom}}% -\label{ref:union-axiom} +\hyperlabel{ref:union-axiom} For any set $A$, there exists a set $B$ whose elements are exactly the members of the members of $A$: @@ -430,7 +430,7 @@ For any set $A$, there exists a set $B$ whose elements are exactly the members \end{axiom} \section{\defined{Union Axiom, Preliminary Form}}% -\label{ref:union-axiom-preliminary-form} +\hyperlabel{ref:union-axiom-preliminary-form} For any sets $a$ and $b$, there is a set whose members are those sets belonging either to $a$ or to $b$ (or both): @@ -446,20 +446,20 @@ For any sets $a$ and $b$, there is a set whose members are those sets belonging \endgroup \chapter{Introduction}% -\label{chap:introduction} +\hyperlabel{chap:introduction} \section{Exercises 1}% -\label{sec:exercises-1} +\hyperlabel{sec:exercises-1} \subsection{\verified{Exercise 1.1}}% -\label{sub:exercise-1.1} +\hyperlabel{sub:exercise-1.1} Which of the following become true when "$\in$" is inserted in place of the blank? Which become true when "$\subseteq$" is inserted? \subsubsection{\verified{Exercise 1.1a}}% -\label{ssub:exercise-1.1a} +\hyperlabel{ssub:exercise-1.1a} $\{\emptyset\} \_\_\_\_ \{\emptyset, \{\emptyset\}\}$. @@ -478,7 +478,7 @@ $\{\emptyset\} \_\_\_\_ \{\emptyset, \{\emptyset\}\}$. \end{proof} \subsubsection{\verified{Exercise 1.1b}}% -\label{ssub:exercise-1.11b} +\hyperlabel{ssub:exercise-1.11b} $\{\emptyset\} \_\_\_\_ \{\emptyset, \{\{\emptyset\}\}\}$. @@ -496,7 +496,7 @@ $\{\emptyset\} \_\_\_\_ \{\emptyset, \{\{\emptyset\}\}\}$. \end{proof} \subsubsection{\verified{Exercise 1.1c}}% -\label{ssub:exercise-1.1c} +\hyperlabel{ssub:exercise-1.1c} $\{\{\emptyset\}\} \_\_\_\_ \{\emptyset, \{\emptyset\}\}$. @@ -514,7 +514,7 @@ $\{\{\emptyset\}\} \_\_\_\_ \{\emptyset, \{\emptyset\}\}$. \end{proof} \subsubsection{\verified{Exercise 1.1d}}% -\label{ssub:exercise-1.1d} +\hyperlabel{ssub:exercise-1.1d} $\{\{\emptyset\}\} \_\_\_\_ \{\emptyset, \{\{\emptyset\}\}\}$. @@ -533,7 +533,7 @@ $\{\{\emptyset\}\} \_\_\_\_ \{\emptyset, \{\{\emptyset\}\}\}$. \end{proof} \subsubsection{\verified{Exercise 1.1e}}% -\label{ssub:exercise-1.1e} +\hyperlabel{ssub:exercise-1.1e} $\{\{\emptyset\}\} \_\_ \{\emptyset, \{\emptyset, \{\emptyset\}\}\}$. @@ -552,7 +552,7 @@ $\{\{\emptyset\}\} \_\_ \{\emptyset, \{\emptyset, \{\emptyset\}\}\}$. \end{proof} \subsection{\verified{Exercise 1.2}}% -\label{sub:exercise-1.2} +\hyperlabel{sub:exercise-1.2} Show that no two of the three sets $\emptyset$, $\{\emptyset\}$, and $\{\{\emptyset\}\}$ are equal to each other. @@ -574,7 +574,7 @@ Show that no two of the three sets $\emptyset$, $\{\emptyset\}$, and \end{proof} \subsection{\verified{Exercise 1.3}}% -\label{sub:exercise-1.3} +\hyperlabel{sub:exercise-1.3} Show that if $B \subseteq C$, then $\powerset{B} \subseteq \powerset{C}$. @@ -593,7 +593,7 @@ Show that if $B \subseteq C$, then $\powerset{B} \subseteq \powerset{C}$. \end{proof} \subsection{\verified{Exercise 1.4}}% -\label{sub:exercise-1.4} +\hyperlabel{sub:exercise-1.4} Assume that $x$ and $y$ are members of a set $B$. Show that $\{\{x\}, \{x, y\}\} \in \powerset{\powerset{B}}.$ @@ -614,7 +614,7 @@ Show that $\{\{x\}, \{x, y\}\} \in \powerset{\powerset{B}}.$ \end{proof} \subsection{\unverified{Exercise 1.5}}% -\label{sub:exercise-1.5} +\hyperlabel{sub:exercise-1.5} Define the rank of a set $c$ to be the least $\alpha$ such that $c \subseteq V_\alpha$. @@ -654,7 +654,7 @@ Compute the rank of \end{proof} \subsection{\unverified{Exercise 1.6}}% -\label{sub:exercise-1.6} +\hyperlabel{sub:exercise-1.6} We have stated that $V_{\alpha + 1} = A \cup \powerset{V_\alpha}$. Prove this at least for $\alpha < 3$. @@ -686,7 +686,7 @@ Prove this at least for $\alpha < 3$. & = (A \cup \powerset{V_{n-1}}) \cup \powerset{V_n} \nonumber \\ & = A \cup (\powerset{V_{n-1}} \cup \powerset{V_n}) - \label{sub:exercise-1.6-eq1} + \hyperlabel{sub:exercise-1.6-eq1} \end{align} But $V_{n-1}$ is a subset of $V_n$. \nameref{sub:exercise-1.3} then implies @@ -702,7 +702,7 @@ Prove this at least for $\alpha < 3$. \end{proof} \subsection{\unverified{Exercise 1.7}}% -\label{sub:exercise-1.7} +\hyperlabel{sub:exercise-1.7} List all the members of $V_3$. List all the members of $V_4$. @@ -744,13 +744,13 @@ List all the members of $V_4$. \end{proof} \chapter{Axioms and Operations}% -\label{chap:axioms-operations} +\hyperlabel{chap:axioms-operations} \section{Axioms}% -\label{sec:axioms} +\hyperlabel{sec:axioms} \subsection{\unverified{Theorem 2A}}% -\label{sub:theorem-2a} +\hyperlabel{sub:theorem-2a} \begin{theorem}[2A] @@ -774,7 +774,7 @@ List all the members of $V_4$. \end{proof} \subsection{\unverified{Theorem 2B}}% -\label{sub:theorem-2b} +\hyperlabel{sub:theorem-2b} \begin{theorem}[2B] @@ -799,10 +799,10 @@ List all the members of $V_4$. \end{proof} \section{Algebra of Sets}% -\label{sec:algebra-sets} +\hyperlabel{sec:algebra-sets} \subsection{\verified{Commutative Laws}}% -\label{sub:commutative-laws} +\hyperlabel{sub:commutative-laws} For any sets $A$ and $B$, \begin{align*} @@ -848,7 +848,7 @@ For any sets $A$ and $B$, \end{proof} \subsection{\verified{Associative Laws}}% -\label{sub:associative-laws} +\hyperlabel{sub:associative-laws} For any sets $A$, $B$ and $C$, \begin{align*} @@ -905,7 +905,7 @@ For any sets $A$, $B$ and $C$, \end{proof} \subsection{\verified{Distributive Laws}}% -\label{sub:distributive-laws} +\hyperlabel{sub:distributive-laws} For any sets $A$, $B$, and $C$, \begin{align*} @@ -961,7 +961,7 @@ For any sets $A$, $B$, and $C$, \end{proof} \subsection{\verified{De Morgan's Laws}}% -\label{sub:de-morgans-laws} +\hyperlabel{sub:de-morgans-laws} For any sets $A$, $B$, and $C$, \begin{align*} @@ -1020,7 +1020,7 @@ For any sets $A$, $B$, and $C$, \subsection{\verified{% Identities Involving \texorpdfstring{$\emptyset$}{the Empty Set}}}% -\label{sub:identitives-involving-empty-set} +\hyperlabel{sub:identitives-involving-empty-set} For any set $A$, \begin{align*} @@ -1089,7 +1089,7 @@ For any set $A$, \end{proof} \subsection{\verified{Monotonicity}}% -\label{sub:monotonicity} +\hyperlabel{sub:monotonicity} For any sets $A$, $B$, and $C$, \begin{align*} @@ -1162,7 +1162,7 @@ For any sets $A$, $B$, and $C$, \end{proof} \subsection{\verified{Anti-monotonicity}}% -\label{sub:anti-monotonicity} +\hyperlabel{sub:anti-monotonicity} For any sets $A$, $B$, and $C$, \begin{align*} @@ -1212,7 +1212,7 @@ For any sets $A$, $B$, and $C$, \end{proof} \subsection{\unverified{General Distributive Laws}}% -\label{sub:general-distributive-laws} +\hyperlabel{sub:general-distributive-laws} For any sets $A$ and $\mathscr{B}$, \begin{align*} @@ -1272,7 +1272,7 @@ For any sets $A$ and $\mathscr{B}$, \end{proof} \subsection{\unverified{General De Morgan's Laws}}% -\label{sub:general-de-morgans-laws} +\hyperlabel{sub:general-de-morgans-laws} For any set $C$ and $\mathscr{A} \neq \emptyset$, \begin{align*} @@ -1333,7 +1333,7 @@ For any set $C$ and $\mathscr{A} \neq \emptyset$, \subsection{\verified{% \texorpdfstring{$\cap$/$-$}{Intersection/Difference} Associativity}}% -\label{sub:intersection-difference-associativity} +\hyperlabel{sub:intersection-difference-associativity} Let $A$, $B$, and $C$ be sets. Then $A \cap (B - C) = (A \cap B) - C$. @@ -1356,7 +1356,7 @@ Then $A \cap (B - C) = (A \cap B) - C$. \end{proof} \subsection{\verified{Nonmembership of Symmetric Difference}} -\label{sub:nonmembership-symmetric-difference} +\hyperlabel{sub:nonmembership-symmetric-difference} Let $A$ and $B$ be sets. $x \not\in A + B$ if and only if either $x \in A \cap B$ or $x \not\in A \cup B$. @@ -1387,10 +1387,10 @@ Let $A$ and $B$ be sets. $x \not\in A + B$ if and only if either \end{proof} \section{Exercises 2}% -\label{sec:exercises-2} +\hyperlabel{sec:exercises-2} \subsection{\verified{Exercise 2.1}}% -\label{sub:exercise-2.1} +\hyperlabel{sub:exercise-2.1} Assume that $A$ is the set of integers divisible by $4$. Similarly assume that $B$ and $C$ are the sets of integers divisible by $9$ and @@ -1407,7 +1407,7 @@ What is in $A \cap B \cap C$? \end{answer} \subsection{\verified{Exercise 2.2}}% -\label{sub:exercise-2.2} +\hyperlabel{sub:exercise-2.2} Give an example of sets $A$ and $B$ for which $\bigcup A = \bigcup B$ but $A \neq B$. @@ -1422,7 +1422,7 @@ Give an example of sets $A$ and $B$ for which $\bigcup A = \bigcup B$ but \end{answer} \subsection{\verified{Exercise 2.3}}% -\label{sub:exercise-2.3} +\hyperlabel{sub:exercise-2.3} Show that every member of a set $A$ is a subset of $\bigcup A$. (This was stated as an example in this section.) @@ -1441,7 +1441,7 @@ Show that every member of a set $A$ is a subset of $\bigcup A$. \end{proof} \subsection{\verified{Exercise 2.4}}% -\label{sub:exercise-2.4} +\hyperlabel{sub:exercise-2.4} Show that if $A \subseteq B$, then $\bigcup A \subseteq \bigcup B$. @@ -1461,7 +1461,7 @@ Show that if $A \subseteq B$, then $\bigcup A \subseteq \bigcup B$. \end{proof} \subsection{\verified{Exercise 2.5}}% -\label{sub:exercise-2.5} +\hyperlabel{sub:exercise-2.5} Assume that every member of $\mathscr{A}$ is a subset of $B$. Show that $\bigcup \mathscr{A} \subseteq B$. @@ -1483,7 +1483,7 @@ Show that $\bigcup \mathscr{A} \subseteq B$. \end{proof} \subsection{\verified{Exercise 2.6a}}% -\label{sub:exercise-2.6a} +\hyperlabel{sub:exercise-2.6a} Show that for any set $A$, $\bigcup \powerset{A} = A$. @@ -1496,7 +1496,7 @@ Show that for any set $A$, $\bigcup \powerset{A} = A$. $A \subseteq \bigcup \powerset{A}$. \paragraph{(i)}% - \label{par:exercise-2.6a-i} + \hyperlabel{par:exercise-2.6a-i} By definition, the \nameref{ref:power-set} of $A$ is the set of all subsets of $A$. @@ -1504,7 +1504,7 @@ Show that for any set $A$, $\bigcup \powerset{A} = A$. By \nameref{sub:exercise-2.5}, $\bigcup \powerset{A} \subseteq A$. \paragraph{(ii)}% - \label{par:exercise-2.6a-ii} + \hyperlabel{par:exercise-2.6a-ii} Let $x \in A$. By definition of the power set of $A$, $\{x\} \in \powerset{A}$. @@ -1523,7 +1523,7 @@ Show that for any set $A$, $\bigcup \powerset{A} = A$. \end{proof} \subsection{\verified{Exercise 2.6b}}% -\label{sub:exercise-2.6b} +\hyperlabel{sub:exercise-2.6b} Show that $A \subseteq \powerset{\bigcup A}$. Under what conditions does equality hold? @@ -1546,13 +1546,13 @@ Under what conditions does equality hold? $A = \powerset{B}$. \paragraph{($\Rightarrow$)}% - \label{par:exercise-2.6b-right} + \hyperlabel{par:exercise-2.6b-right} Suppose $A = \powerset{\bigcup A}$. Then our statement immediately follows by settings $B = \bigcup A$. \paragraph{($\Leftarrow$)}% - \label{par:exercise-2.6b-left} + \hyperlabel{par:exercise-2.6b-left} Suppose there exists some set $B$ such that $A = \powerset{B}$. Therefore @@ -1572,7 +1572,7 @@ Under what conditions does equality hold? \end{proof} \subsection{\verified{Exercise 2.7a}}% -\label{sub:exercise-2.7a} +\hyperlabel{sub:exercise-2.7a} Show that for any sets $A$ and $B$, $$\powerset{A} \cap \powerset{B} = \powerset{(A \cap B)}.$$ @@ -1621,7 +1621,7 @@ Show that for any sets $A$ and $B$, \end{proof} \subsection{\verified{Exercise 2.7b}}% -\label{sub:exercise-2.7b} +\hyperlabel{sub:exercise-2.7b} Show that $\powerset{A} \cup \powerset{B} \subseteq \powerset{(A \cup B)}$. Under what conditions does equality hold? @@ -1653,11 +1653,11 @@ Under what conditions does equality hold? other. \paragraph{($\Rightarrow$)}% - \label{par:exercise-2.7b-right} + \hyperlabel{par:exercise-2.7b-right} Suppose \begin{equation} - \label{sub:exercise-2.7b-eq1} + \hyperlabel{sub:exercise-2.7b-eq1} \powerset{A} \cup \powerset{B} = \powerset{(A \cup B)}. \end{equation} By the definition of the \nameref{ref:power-set}, @@ -1679,7 +1679,7 @@ Under what conditions does equality hold? In other words, $A \subseteq B$ or $B \subseteq A$. \paragraph{($\Leftarrow$)}% - \label{par:exercise-2.7b-left} + \hyperlabel{par:exercise-2.7b-left} WLOG, suppose $A \subseteq B$. Then, by \nameref{sub:exercise-1.3}, $\powerset{A} \subseteq \powerset{B}$. @@ -1700,7 +1700,7 @@ Under what conditions does equality hold? \end{proof} \subsection{\unverified{Exercise 2.8}}% -\label{sub:exercise-2.8} +\hyperlabel{sub:exercise-2.8} Show that there is no set to which every singleton (that is, every set of the form $\{x\}$) belongs. @@ -1719,7 +1719,7 @@ Show that there is no set to which every singleton (that is, every set of the \end{proof} \subsection{\verified{Exercise 2.9}}% -\label{sub:exercise-2.9} +\hyperlabel{sub:exercise-2.9} Give an example of sets $a$ and $B$ for which $a \in B$ but $\powerset{a} \not\in \powerset{B}$. @@ -1740,7 +1740,7 @@ Give an example of sets $a$ and $B$ for which $a \in B$ but \end{answer} \subsection{\verified{Exercise 2.10}}% -\label{sub:exercise-2.10} +\hyperlabel{sub:exercise-2.10} Show that if $a \in B$, then $\powerset{a} \in \powerset{\powerset{\bigcup B}}$. [\textit{Suggestion}: If you need help, look in the Appendix.] @@ -1761,7 +1761,7 @@ Show that if $a \in B$, then $\powerset{a} \in \powerset{\powerset{\bigcup B}}$. \end{proof} \subsection{\verified{Exercise 2.11}}% -\label{sub:exercise-2.11} +\hyperlabel{sub:exercise-2.11} Show that for any sets $A$ and $B$, $$A = (A \cap B) \cup (A - B) \quad\text{and}\quad @@ -1823,7 +1823,7 @@ Show that for any sets $A$ and $B$, \end{proof} \subsection{\verified{Exercise 2.12}}% -\label{sub:exercise-2.12} +\hyperlabel{sub:exercise-2.12} Verify the following identity (one of De Morgan's laws): $$C - (A \cap B) = (C - A) \cup (C - B).$$ @@ -1835,7 +1835,7 @@ Verify the following identity (one of De Morgan's laws): \end{proof} \subsection{\verified{Exercise 2.13}}% -\label{sub:exercise-2.13} +\hyperlabel{sub:exercise-2.13} Show that if $A \subseteq B$, then $C - B \subseteq C - A$. @@ -1846,7 +1846,7 @@ Show that if $A \subseteq B$, then $C - B \subseteq C - A$. \end{proof} \subsection{\verified{Exercise 2.14}}% -\label{sub:exercise-2.14} +\hyperlabel{sub:exercise-2.14} Show by example that for some sets $A$, $B$, and $C$, the set $A - (B - C)$ is different from $(A - B) - C$. @@ -1875,7 +1875,7 @@ Show by example that for some sets $A$, $B$, and $C$, the set $A - (B - C)$ is \end{proof} \subsection{\verified{Exercise 2.15a}}% -\label{sub:exercise-2.15a} +\hyperlabel{sub:exercise-2.15a} Show that $A \cap (B + C) = (A \cap B) + (A \cap C)$. @@ -1912,7 +1912,7 @@ Show that $A \cap (B + C) = (A \cap B) + (A \cap C)$. \end{proof} \subsection{\verified{Exercise 2.15b}}% -\label{sub:exercise-2.15b} +\hyperlabel{sub:exercise-2.15b} Show that $A + (B + C) = (A + B) + C$. @@ -1928,7 +1928,7 @@ Show that $A + (B + C) = (A + B) + C$. \end{enumerate} \paragraph{(i)}% - \label{par:exercise-2.15b-i} + \hyperlabel{par:exercise-2.15b-i} Let $x \in A + (B + C)$. Then $x$ is in $A$ or in $B + C$, but not both. @@ -1964,7 +1964,7 @@ Show that $A + (B + C) = (A + B) + C$. $x \in (A + B) + C$. \paragraph{(ii)}% - \label{par:exercise-2.15b-ii} + \hyperlabel{par:exercise-2.15b-ii} Let $x \in (A + B) + C$. Then $x$ is in $A + B$ or in $C$, but not both. @@ -2008,7 +2008,7 @@ Show that $A + (B + C) = (A + B) + C$. \end{proof} \subsection{\verified{Exercise 2.16}}% -\label{sub:exercise-2.16} +\hyperlabel{sub:exercise-2.16} Simplify: $$[(A \cup B \cup C) \cap (A \cup B)] - [(A \cup (B - C)) \cap A].$$ @@ -2035,7 +2035,7 @@ Simplify: \end{proof} \subsection{\verified{Exercise 2.17}}% -\label{sub:exercise-2.17} +\hyperlabel{sub:exercise-2.17} Show that the following four conditions are equivalent. @@ -2100,7 +2100,7 @@ Show that the following four conditions are equivalent. \end{proof} \subsection{\unverified{Exercise 2.18}}% -\label{sub:exercise-2.18} +\hyperlabel{sub:exercise-2.18} Assume that $A$ and $B$ are subsets of $S$. List all of the different sets that can be made from these three by use of the @@ -2132,7 +2132,7 @@ List all of the different sets that can be made from these three by use of the \end{proof} \subsection{\verified{Exercise 2.19}}% -\label{sub:exercise-2.19} +\hyperlabel{sub:exercise-2.19} Is $\powerset{(A - B)}$ always equal to $\powerset{A} - \powerset{B}$? Is it ever equal to $\powerset{A} - \powerset{B}$? @@ -2147,7 +2147,7 @@ Is it ever equal to $\powerset{A} - \powerset{B}$? $\emptyset \not\in \powerset{A} - \powerset{B}$. \paragraph{(i)}% - \label{par:exercise-2.19-i} + \hyperlabel{par:exercise-2.19-i} By definition of the \nameref{ref:power-set}, $$\powerset{(A - B)} = \{ x \mid x \subseteq A - B \}.$$ @@ -2168,7 +2168,7 @@ Is it ever equal to $\powerset{A} - \powerset{B}$? \end{proof} \subsection{\verified{Exercise 2.20}}% -\label{sub:exercise-2.20} +\hyperlabel{sub:exercise-2.20} Let $A$, $B$, and $C$ be sets such that $A \cup B = A \cup C$ and $A \cap B = A \cap C$. @@ -2227,7 +2227,7 @@ Show that $B = C$. \end{proof} \subsection{\verified{Exercise 2.21}}% -\label{sub:exercise-2.21} +\hyperlabel{sub:exercise-2.21} Show that $\bigcup (A \cup B) = \bigcup A \cup \bigcup B$. @@ -2265,7 +2265,7 @@ Show that $\bigcup (A \cup B) = \bigcup A \cup \bigcup B$. \end{proof} \subsection{\verified{Exercise 2.22}}% -\label{sub:exercise-2.22} +\hyperlabel{sub:exercise-2.22} Show that if $A$ and $B$ are nonempty sets, then $\bigcap (A \cup B) = \bigcap A \cap \bigcap B$. @@ -2279,7 +2279,7 @@ Show that if $A$ and $B$ are nonempty sets, then By the \nameref{ref:extensionality-axiom}, the specified equality holds if and only if for all sets $x$, \begin{equation} - \label{sub:exercise-2.22-eq1} + \hyperlabel{sub:exercise-2.22-eq1} x \in \bigcap (A \cup B) \iff x \in \bigcap A \cap \bigcap B. \end{equation} We prove both directions of this biconditional. @@ -2305,7 +2305,7 @@ Show that if $A$ and $B$ are nonempty sets, then \end{proof} \subsection{\unverified{Exercise 2.23}}% -\label{sub:exercise-2.23} +\hyperlabel{sub:exercise-2.23} Show that if $\mathscr{B}$ is nonempty, then $A \cup \bigcap \mathscr{B} = \bigcap\; \{A \cup X \mid X \in \mathscr{B} \}$. @@ -2317,7 +2317,7 @@ Show that if $\mathscr{B}$ is nonempty, then \end{proof} \subsection{\verified{Exercise 2.24a}}% -\label{sub:exercise-2.24a} +\hyperlabel{sub:exercise-2.24a} Show that if $\mathscr{A}$ is nonempty, then $\powerset{\bigcap\mathscr{A}} = @@ -2356,11 +2356,11 @@ Show that if $\mathscr{A}$ is nonempty, then \end{proof} \subsection{\verified{Exercise 2.24b}}% -\label{sub:exercise-2.24b} +\hyperlabel{sub:exercise-2.24b} Show that \begin{equation} - \label{sub:exercise-2.24b-eq1} + \hyperlabel{sub:exercise-2.24b-eq1} \bigcup\; \{ \powerset{X} \mid X \in \mathscr{A} \} \subseteq \powerset{\bigcup\mathscr{A}}. \end{equation} @@ -2419,7 +2419,7 @@ Under what conditions does equality hold? \end{proof} \subsection{\verified{Exercise 2.25}}% -\label{sub:exercise-2.25} +\hyperlabel{sub:exercise-2.25} Is $A \cup \bigcup \mathscr{B}$ always the same as $\bigcup\;\{ A \cup X \mid X \in \mathscr{B} \}$? @@ -2432,7 +2432,7 @@ If not, then under what conditions does equality hold? We prove that \begin{equation} - \label{sub:exercise-2.25-eq1} + \hyperlabel{sub:exercise-2.25-eq1} A \cup \bigcup \mathscr{B} = \bigcup\;\{ A \cup X \mid X \in \mathscr{B} \} \end{equation} @@ -2496,19 +2496,19 @@ If not, then under what conditions does equality hold? \end{proof} \chapter{Relations and Functions}% -\label{chap:relations-functions} +\hyperlabel{chap:relations-functions} \section{Ordered Pairs}% -\label{sec:ordered-pairs} +\hyperlabel{sec:ordered-pairs} \subsection{\verified{Theorem 3A}}% -\label{sub:theorem-3a} +\hyperlabel{sub:theorem-3a} \begin{theorem}[3A] For any sets $x$, $y$, $u$, and $v$, \begin{equation} - \label{sub:theorem-3a-eq1} + \hyperlabel{sub:theorem-3a-eq1} \pair{u, v} = \pair{x, y} \iff u = x \land v = y. \end{equation} @@ -2530,7 +2530,7 @@ If not, then under what conditions does equality hold? Suppose $\pair{u, v} = \pair{x, y}$. Then, by definition of an \nameref{ref:ordered-pair}, \begin{equation} - \label{sub:theorem-3a-eq2} + \hyperlabel{sub:theorem-3a-eq2} \{\{u\}, \{u, v\}\} = \{\{x\}, \{x, y\}\}. \end{equation} By the \nameref{ref:extensionality-axiom}, it follows @@ -2578,7 +2578,7 @@ If not, then under what conditions does equality hold? \end{proof} \subsection{\verified{Lemma 3B}}% -\label{sub:lemma-3b} +\hyperlabel{sub:lemma-3b} \begin{theorem}[3B] @@ -2602,7 +2602,7 @@ If not, then under what conditions does equality hold? \end{proof} \subsection{\verified{Corollary 3C}}% -\label{sub:corollary-3c} +\hyperlabel{sub:corollary-3c} \begin{theorem}[3C] @@ -2635,10 +2635,10 @@ If not, then under what conditions does equality hold? \end{proof} \section{Relations}% -\label{sec:relations} +\hyperlabel{sec:relations} \subsection{\verified{Theorem 3D}}% -\label{sub:theorem-3d} +\hyperlabel{sub:theorem-3d} \begin{theorem}[3D] @@ -2663,10 +2663,10 @@ If not, then under what conditions does equality hold? \end{proof} \section{Functions}% -\label{sec:functions} +\hyperlabel{sec:functions} \subsection{\verified{Theorem 3E}}% -\label{sub:theorem-3e} +\hyperlabel{sub:theorem-3e} \begin{theorem}[3E] @@ -2724,7 +2724,7 @@ If not, then under what conditions does equality hold? \end{proof} \subsection{\verified{Theorem 3F}}% -\label{sub:theorem-3f} +\hyperlabel{sub:theorem-3f} \begin{theorem}[3F] @@ -2748,7 +2748,7 @@ If not, then under what conditions does equality hold? single-rooted. \paragraph{(i)}% - \label{par:theorem-3f-i} + \hyperlabel{par:theorem-3f-i} Let $F$ be any set. @@ -2794,7 +2794,7 @@ If not, then under what conditions does equality hold? \end{proof} \subsection{\verified{Lemma 1}}% -\label{sub:lemma-1} +\hyperlabel{sub:lemma-1} For any one-to-one function $F$, $F^{-1}$ is also one-to-one. @@ -2806,7 +2806,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. We prove that (i) $F^{-1}$ is a function and (ii) $F^{-1}$ is single-rooted. \paragraph{(i)}% - \label{par:lemma-1-i} + \hyperlabel{par:lemma-1-i} By hypothesis, $F$ is one-to-one. This means it is single-rooted, i.e. for all $x \in \ran{F}$, there exists @@ -2818,7 +2818,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. Thus $F^{-1}$ is a function. \paragraph{(ii)}% - \label{par:lemma-1-ii} + \hyperlabel{par:lemma-1-ii} By hypothesis, $F$ is single-valued. That is, for all $x \in \dom{F}$, there exists exactly one $y$ such that @@ -2837,7 +2837,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\verified{Theorem 3G}}% -\label{sub:theorem-3g} +\hyperlabel{sub:theorem-3g} \begin{theorem}[3G] @@ -2872,14 +2872,14 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\verified{Theorem 3H}}% -\label{sub:theorem-3h} +\hyperlabel{sub:theorem-3h} \begin{theorem}[3H] Assume that $F$ and $G$ are functions. Then $F \circ G$ is a function, its domain is \begin{equation} - \label{sub:theorem-3h-eq1} + \hyperlabel{sub:theorem-3h-eq1} \{x \in \dom{G} \mid G(x) \in \dom{F}\}, \end{equation} and for $x$ in its domain, $(F \circ G)(x) = F(G(x))$. @@ -2899,7 +2899,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. Let $F$ and $G$ be \nameref{ref:function}s. By definition of the \nameref{ref:composition} of $F$ and $G$, \begin{equation} - \label{sub:theorem-3h-eq2} + \hyperlabel{sub:theorem-3h-eq2} F \circ G = \{\pair{u, v} \mid \exists t(uGt \land tFv)\}. \end{equation} By construction, $F \circ G$ is a relation. @@ -2910,7 +2910,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \eqref{sub:theorem-3h-eq1}, and (ii) $(F \circ G)(x) = F(G(x))$. \paragraph{(i)}% - \label{par:theorem-3h-i} + \hyperlabel{par:theorem-3h-i} By \eqref{sub:theorem-3h-eq2}, there exists some $t$ such that $\pair{x, t} \in G$ and $\pair{t, y} \in F$. @@ -2934,7 +2934,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\verified{Theorem 3I}}% -\label{sub:theorem-3i} +\hyperlabel{sub:theorem-3i} \begin{theorem}[3I] @@ -2962,7 +2962,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\pending{Theorem 3J}}% -\label{sub:theorem-3j} +\hyperlabel{sub:theorem-3j} \begin{theorem}[3J] @@ -3047,19 +3047,19 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\verified{Theorem 3K(a)}}% -\label{sub:theorem-3k-a} +\hyperlabel{sub:theorem-3k-a} \begin{theorem}[3K(a)] The following hold for any sets. ($F$ need not be a function.) The image of a union is the union of the images: \begin{equation} - \label{sub:theorem-3k-a-eq1} + \hyperlabel{sub:theorem-3k-a-eq1} \img{F}{A \cup B} = \img{F}{A} \cup \img{F}{B} \end{equation} and \begin{equation} - \label{sub:theorem-3k-a-eq2} + \hyperlabel{sub:theorem-3k-a-eq2} \img{F}{\bigcup{\mathscr{A}}} = \bigcup\;\{\img{F}{A} \mid A \in \mathscr{A}\}. \end{equation} @@ -3118,19 +3118,19 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\verified{Theorem 3K(b)}}% -\label{sub:theorem-3k-b} +\hyperlabel{sub:theorem-3k-b} \begin{theorem}[3K(b)] The following hold for any sets. ($F$ need not be a function.) The image of an intersection is included in the intersection of the images: \begin{equation} - \label{sub:theorem-3k-b-eq1} + \hyperlabel{sub:theorem-3k-b-eq1} \img{F}{A \cap B} \subseteq \img{F}{A} \cap \img{F}{B} \end{equation} and \begin{equation} - \label{sub:theorem-3k-b-eq2} + \hyperlabel{sub:theorem-3k-b-eq2} \img{F}{\bigcap\mathscr{A}} \subseteq \bigcap\;\{\img{F}{A} \mid A \in \mathscr{A}\}. \end{equation} @@ -3158,7 +3158,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. under equality. \paragraph{(i)}% - \label{par:theorem-3k-b-i} + \hyperlabel{par:theorem-3k-b-i} Let $v \in \img{F}{A \cap B}$. By definition of the \nameref{ref:image} of a set, @@ -3167,7 +3167,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. Therefore $v \in \img{F}{A} \cap \img{F}{B}$. \paragraph{(ii)}% - \label{par:theorem-3k-b-ii} + \hyperlabel{par:theorem-3k-b-ii} Let $v \in \img{F}{\bigcap{\mathscr{A}}}$. By definition of the \nameref{ref:image} of a set, @@ -3211,14 +3211,14 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\verified{Theorem 3K(c)}}% -\label{sub:theorem-3k-c} +\hyperlabel{sub:theorem-3k-c} \begin{theorem}[3K(c)] The following hold for any sets. ($F$ need not be a function.) The image of a difference includes the difference of the images: \begin{equation} - \label{sub:theorem-3k-c-eq1} + \hyperlabel{sub:theorem-3k-c-eq1} \img{F}{A} - \img{F}{B} \subseteq \img{F}{A - B}. \end{equation} Equality holds if $F$ is single-rooted. @@ -3239,7 +3239,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. if $F$ is single-rooted. \paragraph{(i)}% - \label{par:theorem-3k-c-i} + \hyperlabel{par:theorem-3k-c-i} Let $v \in \img{F}{A} - \img{F}{B}$. By definition of the difference of two sets, @@ -3269,7 +3269,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\verified{Corollary 3L}}% -\label{sub:corollary-3l} +\hyperlabel{sub:corollary-3l} \begin{theorem}[3L] @@ -3277,13 +3277,13 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \begin{align} \img{G^{-1}}{\bigcup{\mathscr{A}}} & = \bigcup\;\{\img{G^{-1}}{A} \mid A \in \mathscr{A}\}, - \label{sub:corollary-3l-eq1} \\ + \hyperlabel{sub:corollary-3l-eq1} \\ \img{G^{-1}}{\bigcap{\mathscr{A}}} & = \bigcap\;\{\img{G^{-1}}{A} \mid A \in \mathscr{A}\} \text{ for } \mathscr{A} \neq \emptyset, - \label{sub:corollary-3l-eq2} \\ + \hyperlabel{sub:corollary-3l-eq2} \\ \img{G^{-1}}{A - B} & = \img{G^{-1}}{A} - \img{G^{-1}}{B}. - \label{sub:corollary-3l-eq3} + \hyperlabel{sub:corollary-3l-eq3} \end{align} \end{theorem} @@ -3309,10 +3309,10 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \section{Equivalence Relations}% -\label{sec:equivalence-relations} +\hyperlabel{sec:equivalence-relations} \subsection{\verified{Theorem 3M}}% -\label{sub:theorem-3m} +\hyperlabel{sub:theorem-3m} \begin{theorem}[3M] @@ -3348,7 +3348,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\verified{Lemma 3N}}% -\label{sub:lemma-3n} +\hyperlabel{sub:lemma-3n} \begin{lemma}[3N] @@ -3397,7 +3397,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\verified{Theorem 3P}}% -\label{sub:theorem-3p} +\hyperlabel{sub:theorem-3p} \begin{theorem}[3P] @@ -3447,7 +3447,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \subsection{\pending{Theorem 3Q}}% -\label{sub:theorem-3q} +\hyperlabel{sub:theorem-3q} \begin{theorem}[3Q] @@ -3456,7 +3456,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. If $F$ is compatible with $R$, then there exists a unique $\hat{F} \colon A / R \rightarrow A / R$ such that \begin{equation} - \label{sub:theorem-3q-eq1} + \hyperlabel{sub:theorem-3q-eq1} \hat{F}([x]_R) = [F(x)]_R \quad\text{for all } x \text{ in } A. \end{equation} If $F$ is not compatible with $R$, then no such $\hat{F}$ exists. @@ -3499,10 +3499,10 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} \section{Exercises 3}% -\label{sec:exercises-3} +\hyperlabel{sec:exercises-3} \subsection{\verified{Exercise 3.1}}% -\label{sub:exercise-3.1} +\hyperlabel{sub:exercise-3.1} Suppose that we attempted to generalize the Kuratowski definitions of ordered pairs to ordered triples by defining @@ -3538,7 +3538,7 @@ Show that this definition is unsuccessful by giving examples of objects \end{proof} \subsection{\verified{Exercise 3.2a}}% -\label{sub:exercise-3.2a} +\hyperlabel{sub:exercise-3.2a} Show that $A \times (B \cup C) = (A \times B) \cup (A \times C)$. @@ -3564,7 +3564,7 @@ Show that $A \times (B \cup C) = (A \times B) \cup (A \times C)$. \end{proof} \subsection{\verified{Exercise 3.2b}}% -\label{sub:exercise-3.2b} +\hyperlabel{sub:exercise-3.2b} Show that if $A \times B = A \times C$ and $A \neq \emptyset$, then $B = C$. @@ -3577,9 +3577,9 @@ Show that if $A \times B = A \times C$ and $A \neq \emptyset$, then $B = C$. By \nameref{sub:corollary-3c}, \begin{align} A \times B & = \{ \pair{x, y} \mid x \in A \land y \in B \} - & \label{sub:exercise-3.2b-eq1} \\ + & \hyperlabel{sub:exercise-3.2b-eq1} \\ A \times C & = \{ \pair{x, y} \mid x \in A \land y \in C \}. - & \label{sub:exercise-3.2b-eq2} + & \hyperlabel{sub:exercise-3.2b-eq2} \end{align} There are two cases to consider: @@ -3609,7 +3609,7 @@ Show that if $A \times B = A \times C$ and $A \neq \emptyset$, then $B = C$. \end{proof} \subsection{\verified{Exercise 3.3}}% -\label{sub:exercise-3.3} +\hyperlabel{sub:exercise-3.3} Show that $A \times \bigcup \mathscr{B} = \bigcup\;\{ A \times X \mid X \in \mathscr{B} \}$. @@ -3635,7 +3635,7 @@ Show that $A \times \bigcup \mathscr{B} = \end{proof} \subsection{\unverified{Exercise 3.4}}% -\label{sub:exercise-3.4} +\hyperlabel{sub:exercise-3.4} Show that there is no set to which every ordered pair belongs. @@ -3654,12 +3654,12 @@ Show that there is no set to which every ordered pair belongs. \end{proof} \subsection{\verified{Exercise 3.5a}}% -\label{sub:exercise-3.5a} +\hyperlabel{sub:exercise-3.5a} Assume that $A$ and $B$ are given sets, and show that there exists a set $C$ such that for any $y$, \begin{equation} - \label{sub:exercise-3.5a-eq1} + \hyperlabel{sub:exercise-3.5a-eq1} y \in C \iff y = \{x\} \times B \text{ for some } x \text{ in } A. \end{equation} In other words, show that $\{\{x\} \times B \mid x \in A\}$ is a set. @@ -3709,7 +3709,7 @@ In other words, show that $\{\{x\} \times B \mid x \in A\}$ is a set. \end{proof} \subsection{\verified{Exercise 3.5b}}% -\label{sub:exercise-3.5b} +\hyperlabel{sub:exercise-3.5b} With $A$, $B$, and $C$ as above, show that $A \times B = \bigcup C$. @@ -3721,7 +3721,7 @@ With $A$, $B$, and $C$ as above, show that $A \times B = \bigcup C$. Let $A$ and $B$ be arbitrary sets. We want to show that \begin{equation} - \label{sub:exercise-3.5b-eq1} + \hyperlabel{sub:exercise-3.5b-eq1} A \times B = \bigcup\; \{\{x\} \times B \mid x \in A\}. \end{equation} The left-hand side of \eqref{sub:exercise-3.5b-eq1} is a set by virtue of @@ -3759,7 +3759,7 @@ With $A$, $B$, and $C$ as above, show that $A \times B = \bigcup C$. \end{proof} \subsection{\verified{Exercise 3.6}}% -\label{sub:exercise-3.6} +\hyperlabel{sub:exercise-3.6} Show that a set $A$ is a relation iff $A \subseteq \dom{A} \times \ran{A}$. @@ -3793,7 +3793,7 @@ Show that a set $A$ is a relation iff $A \subseteq \dom{A} \times \ran{A}$. \end{proof} \subsection{\verified{Exercise 3.7}}% -\label{sub:exercise-3.7} +\hyperlabel{sub:exercise-3.7} Show that if $R$ is a relation, then $\fld{R} = \bigcup\bigcup R$. @@ -3807,7 +3807,7 @@ Show that if $R$ is a relation, then $\fld{R} = \bigcup\bigcup R$. $\bigcup\bigcup R \subseteq \fld{R}$. \paragraph{(i)}% - \label{par:exercise-3.7-i} + \hyperlabel{par:exercise-3.7-i} Let $x \in \fld{R} = \dom{R} \cup \ran{R}$. That is, $x \in \dom{R}$ or $x \in \ran{R}$. @@ -3821,7 +3821,7 @@ Show that if $R$ is a relation, then $\fld{R} = \bigcup\bigcup R$. Then $\{t, x\} \in \bigcup R$ and $x \in \bigcup\bigcup R$. \paragraph{(ii)}% - \label{par:exercise-3.7-ii} + \hyperlabel{par:exercise-3.7-ii} Let $t \in \bigcup\bigcup R$. Then there exists some member $T \in \bigcup R$ such that $t \in T$. @@ -3840,16 +3840,16 @@ Show that if $R$ is a relation, then $\fld{R} = \bigcup\bigcup R$. \end{proof} \subsection{\verified{Exercise 3.8}}% -\label{sub:exercise-3.8} +\hyperlabel{sub:exercise-3.8} Show that for any set $\mathscr{A}$: \begin{align} \dom{\bigcup{\mathscr{A}}} & = \bigcup\;\{ \dom{R} \mid R \in \mathscr{A} \}, - & \label{sub:exercise-3.8-eq1} \\ + & \hyperlabel{sub:exercise-3.8-eq1} \\ \ran{\bigcup{\mathscr{A}}} & = \bigcup\;\{ \ran{R} \mid R \in \mathscr{A} \}. - & \label{sub:exercise-3.8-eq2} + & \hyperlabel{sub:exercise-3.8-eq2} \end{align} \begin{proof} @@ -3900,7 +3900,7 @@ Show that for any set $\mathscr{A}$: \end{proof} \subsection{\verified{Exercise 3.9}}% -\label{sub:exercise-3.9} +\hyperlabel{sub:exercise-3.9} Discuss the result of replacing the union operation by the intersection operation in the preceding problem. @@ -3920,10 +3920,10 @@ Discuss the result of replacing the union operation by the intersection \begin{align} \dom{\bigcap{\mathscr{A}}} & \subseteq \bigcap\;\{ \dom{R} \mid R \in \mathscr{A} \}, - & \label{sub:exercise-3.9-eq1} \\ + & \hyperlabel{sub:exercise-3.9-eq1} \\ \ran{\bigcap{\mathscr{A}}} & \subseteq \bigcap\;\{ \ran{R} \mid R \in \mathscr{A} \}. - & \label{sub:exercise-3.9-eq2} + & \hyperlabel{sub:exercise-3.9-eq2} \end{align} We prove (i) \eqref{sub:exercise-3.9-eq1} and then (ii) @@ -3962,7 +3962,7 @@ Discuss the result of replacing the union operation by the intersection \end{answer} \subsection{\unverified{Exercise 3.10}}% -\label{sub:exercise-3.10} +\hyperlabel{sub:exercise-3.10} Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive integer $m$ less than $4$. @@ -3974,9 +3974,9 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive \begin{align} \pair{x_1, x_2, x_3, x_4} & = \pair{\pair{x_1, x_2, x_3}, x_4} - & \label{sub:exercise-7.10-eq1} \\ + & \hyperlabel{sub:exercise-7.10-eq1} \\ & = \pair{\pair{\pair{x_1, x_2}, x_3}, x_4} - & \label{sub:exercise-7.10-eq2} + & \hyperlabel{sub:exercise-7.10-eq2} \end{align} Here \eqref{sub:exercise-7.10-eq1} is an equivalent ordered $2$-tuple and \eqref{sub:exercise-7.10-eq2} is an equivalent ordered $3$-tuple. @@ -3986,7 +3986,7 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive \end{answer} \subsection{\verified{Exercise 3.11}}% -\label{sub:exercise-3.11} +\hyperlabel{sub:exercise-3.11} Prove the following version (for functions) of the extensionality principle: Assume that $F$ and $G$ are functions, $\dom{F} = \dom{G}$, and @@ -4012,7 +4012,7 @@ Then $F = G$. \end{proof} \subsection{\verified{Exercise 3.12}}% -\label{sub:exercise-3.12} +\hyperlabel{sub:exercise-3.12} Assume that $f$ and $g$ are functions and show that $$f \subseteq g \iff \dom{f} \subseteq \dom{g} \land @@ -4048,7 +4048,7 @@ Assume that $f$ and $g$ are functions and show that \end{proof} \subsection{\verified{Exercise 3.13}}% -\label{sub:exercise-3.13} +\hyperlabel{sub:exercise-3.13} Assume that $f$ and $g$ are functions with $f \subseteq g$ and $\dom{g} \subseteq \dom{f}$. @@ -4070,7 +4070,7 @@ Show that $f = g$. \end{proof} \subsection{\verified{Exercise 3.14}}% -\label{sub:exercise-3.14} +\hyperlabel{sub:exercise-3.14} Assume that $f$ and $g$ are functions. @@ -4137,7 +4137,7 @@ Assume that $f$ and $g$ are functions. \end{proof} \subsection{\verified{Exercise 3.15}}% -\label{sub:exercise-3.15} +\hyperlabel{sub:exercise-3.15} Let $\mathscr{A}$ be a set of functions such that for any $f$ and $g$ in $\mathscr{A}$, either $f \subseteq g$ or $g \subseteq f$. @@ -4183,7 +4183,7 @@ Show that $\bigcup{\mathscr{A}}$ is a function. \end{proof} \subsection{\unverified{Exercise 3.16}}% -\label{sub:exercise-3.16} +\hyperlabel{sub:exercise-3.16} Show that there is no set to which every function belongs. @@ -4199,7 +4199,7 @@ Show that there is no set to which every function belongs. \end{proof} \subsection{\verified{Exercise 3.17}}% -\label{sub:exercise-3.17} +\hyperlabel{sub:exercise-3.17} Show that the composition of two single-rooted sets is again single-rooted. Conclude that the composition of two one-to-one functions is again one-to-one. @@ -4218,7 +4218,7 @@ Conclude that the composition of two one-to-one functions is again one-to-one. Consider $F \circ G$. By definition of the \nameref{ref:composition} of sets, \begin{equation} - \label{sub:exercise-3.17-eq1} + \hyperlabel{sub:exercise-3.17-eq1} F \circ G = \{\pair{u, v} \mid \exists t(uGt \land tFv)\}. \end{equation} Consider any $v \in \ran{(F \circ G)}$. @@ -4244,7 +4244,7 @@ Conclude that the composition of two one-to-one functions is again one-to-one. \end{proof} \subsection{\verified{Exercise 3.18}}% -\label{sub:exercise-3.18} +\hyperlabel{sub:exercise-3.18} Let $R$ be the set $$\{ \pair{0, 1}, \pair{0, 2}, \pair{0, 3}, @@ -4282,7 +4282,7 @@ Evaluate the following: $R \circ R$, $R \restriction \{1\}$, \end{proof} \subsection{\verified{Exercise 3.19}}% -\label{sub:exercise-3.19} +\hyperlabel{sub:exercise-3.19} Let $$A = \{ \pair{\emptyset, \{\emptyset, \{\emptyset\}\}}, @@ -4351,7 +4351,7 @@ Evaluate each of the following: $A(\emptyset)$, $\img{A}{\emptyset}$, \end{proof} \subsection{\verified{Exercise 3.20}}% -\label{sub:exercise-3.20} +\hyperlabel{sub:exercise-3.20} Show that $F \restriction A = F \cap (A \times \ran{F})$. @@ -4377,7 +4377,7 @@ Show that $F \restriction A = F \cap (A \times \ran{F})$. \end{proof} \subsection{\verified{Exercise 3.21}}% -\label{sub:exercise-3.21} +\hyperlabel{sub:exercise-3.21} Show that $(R \circ S) \circ T = R \circ (S \circ T)$. @@ -4408,7 +4408,7 @@ Show that $(R \circ S) \circ T = R \circ (S \circ T)$. \end{proof} \subsection{\verified{Exercise 3.22}}% -\label{sub:exercise-3.22} +\hyperlabel{sub:exercise-3.22} Show that the following are correct for any sets. @@ -4423,13 +4423,13 @@ Show that the following are correct for any sets. \statementpadding - \lean*{Bookshelf/Enderton/Set/Relation} + \lean*{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_22\_a} - \lean*{Bookshelf/Enderton/Set/Relation} + \lean*{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_22\_b} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_22\_c} Let $A$, $B$, $F$, $G$, and $Q$ be arbitrary sets. @@ -4481,7 +4481,7 @@ Show that the following are correct for any sets. \end{proof} \subsection{\verified{Exercise 3.23}}% -\label{sub:exercise-3.23} +\hyperlabel{sub:exercise-3.23} Let $I_A$ be the identity function on the set $A$. Show that for any sets $B$ and $C$, @@ -4492,10 +4492,10 @@ Show that for any sets $B$ and $C$, \statementpadding - \lean*{Bookshelf/Enderton/Set/Relation} + \lean*{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_23\_i} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_23\_ii} Let $I_A$ be the identity function on the set $A$. @@ -4543,14 +4543,14 @@ Show that for any sets $B$ and $C$, \end{proof} \subsection{\verified{Exercise 3.24}}% -\label{sub:exercise-3.24} +\hyperlabel{sub:exercise-3.24} Show that for a function $F$, $\img{F^{-1}}{A} = \{x \in \dom{F} \mid F(x) \in A\}$. \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_24} Let $F$ be a function. @@ -4567,7 +4567,7 @@ Show that for a function $F$, \end{proof} \subsection{\verified{Exercise 3.25}}% -\label{sub:exercise-3.25} +\hyperlabel{sub:exercise-3.25} \begin{enumerate}[(a)] \item Assume that $G$ is a one-to-one function. @@ -4581,14 +4581,14 @@ Show that for a function $F$, \statementpadding - \lean*{Bookshelf/Enderton/Set/Relation} + \lean*{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_25\_b} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_25\_a} \paragraph{(b)}% - \label{par:exercise-3.25-b} + \hyperlabel{par:exercise-3.25-b} Let $G$ be an arbitrary function. We show that $G \circ G^{-1} \subseteq I_{\ran{G}}$ and that @@ -4626,7 +4626,7 @@ Show that for a function $F$, \end{proof} \subsection{\verified{Exercise 3.26}}% -\label{sub:exercise-3.26} +\hyperlabel{sub:exercise-3.26} Prove the second halves of parts (a) and (b) of Theorem 3K. @@ -4638,14 +4638,14 @@ Prove the second halves of parts (a) and (b) of Theorem 3K. \end{proof} \subsection{\verified{Exercise 3.27}}% -\label{sub:exercise-3.27} +\hyperlabel{sub:exercise-3.27} Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$. ($F$ and $G$ need not be functions.) \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_27} Let $F$ and $G$ be arbitrary sets. @@ -4682,7 +4682,7 @@ Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$. \end{proof} \subsection{\verified{Exercise 3.28}}% -\label{sub:exercise-3.28} +\hyperlabel{sub:exercise-3.28} Assume that $f$ is a one-to-one function from $A$ into $B$, and that $G$ is the function with $\dom{G} = \powerset{A}$ defined by the equation @@ -4691,7 +4691,7 @@ Show that $G$ maps $\powerset{A}$ one-to-one into $\powerset{B}$. \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_28} By construction, $\dom{G} = \powerset{A}$. @@ -4715,12 +4715,12 @@ Show that $G$ maps $\powerset{A}$ one-to-one into $\powerset{B}$. \end{proof} \subsection{\verified{Exercise 3.29}}% -\label{sub:exercise-3.29} +\hyperlabel{sub:exercise-3.29} Assume that $f \colon A \rightarrow B$ and define a function $G \colon B \rightarrow \powerset{A}$ by \begin{equation} - \label{sub:exercise-3.29-eq1} + \hyperlabel{sub:exercise-3.29-eq1} G(b) = \{x \in A \mid f(x) = b\}. \end{equation} Show that if $f$ maps $A$ \textit{onto} $B$, then $G$ is one-to-one. @@ -4728,8 +4728,8 @@ Does the converse hold? \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} - {Enderton.Set.Chapter\_3.exercise\_3\_29} + \lean{Bookshelf/Enderton/Set/Chapter\_3} + {Enderton.Set.Chapter\_3.exercise\_3\_39} Let $f \colon A \rightarrow B$ such that $f$ maps $A$ onto $B$. Define $G \colon B \rightarrow \powerset{A}$ by @@ -4769,7 +4769,7 @@ Does the converse hold? \end{proof} \subsection{\sorry{Exercise 3.30}}% -\label{sub:exercise-3.30} +\hyperlabel{sub:exercise-3.30} Assume that $F \colon \powerset{A} \rightarrow \powerset{A}$ and that $F$ has the monotonicity property: @@ -4789,7 +4789,7 @@ Define \end{proof} \subsection{\unverified{Exercise 3.31}}% -\label{sub:exercise-3.31} +\hyperlabel{sub:exercise-3.31} Show that from the first form of the axiom of choice we can prove the second form, and conversely. @@ -4849,13 +4849,13 @@ Show that from the first form of the axiom of choice we can prove the second \end{proof} \subsection{\verified{Exercise 3.32a}}% -\label{sub:exercise-3.32-a} +\hyperlabel{sub:exercise-3.32-a} Show that $R$ is symmetric iff $R^{-1} \subseteq R$. \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_32\_a} \paragraph{($\Rightarrow$)}% @@ -4880,13 +4880,13 @@ Show that $R$ is symmetric iff $R^{-1} \subseteq R$. \end{proof} \subsection{\verified{Exercise 3.32b}}% -\label{sub:exercise-3.32-b} +\hyperlabel{sub:exercise-3.32-b} Show that $R$ is transitive iff $R \circ R \subseteq R$. \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_32\_b} \paragraph{($\Rightarrow$)}% @@ -4911,13 +4911,13 @@ Show that $R$ is transitive iff $R \circ R \subseteq R$. \end{proof} \subsection{\verified{Exercise 3.33}}% -\label{sub:exercise-3.33} +\hyperlabel{sub:exercise-3.33} Show that $R$ is a symmetric and transitive relation iff $R = R^{-1} \circ R$. \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_33} By definition of the \nameref{ref:inverse} and \nameref{ref:composition} @@ -4927,7 +4927,7 @@ Show that $R$ is a symmetric and transitive relation iff $R = R^{-1} \circ R$. & = \{ (u, v) \mid \exists t(uRt \land tR^{-1}v) \} \nonumber \\ & = \{ (u, v) \mid \exists t(uRt \land vRt) \}. - \label{sub:exercise-3.33-eq1} + \hyperlabel{sub:exercise-3.33-eq1} \end{align} \paragraph{($\Rightarrow$)}% @@ -4960,7 +4960,7 @@ Show that $R$ is a symmetric and transitive relation iff $R = R^{-1} \circ R$. We prove that (i) $R$ is symmetric and (ii) $R$ is transitive. \subparagraph{(i)}% - \label{spar:exercise-3.33-i} + \hyperlabel{spar:exercise-3.33-i} First we note that $R$ is equal to its inverse: \begin{align} @@ -4970,7 +4970,7 @@ Show that $R$ is a symmetric and transitive relation iff $R = R^{-1} \circ R$. & \textref{sub:theorem-3i} \nonumber \\ & = R^{-1} \circ R & \textref{sub:theorem-3e} \nonumber \\ - & = R \label{sub:exercise-3.33-eq2}. + & = R \hyperlabel{sub:exercise-3.33-eq2}. \end{align} Now let $\pair{x, y} \in R$. By \eqref{sub:exercise-3.33-eq2} $\pair{x, y} \in R^{-1}$. @@ -4991,7 +4991,7 @@ Show that $R$ is a symmetric and transitive relation iff $R = R^{-1} \circ R$. \end{proof} \subsection{\verified{Exercise 3.34}}% -\label{sub:exercise-3.34} +\hyperlabel{sub:exercise-3.34} Assume that $\mathscr{A}$ is a nonempty set, every member of which is a transitive relation. @@ -5005,10 +5005,10 @@ Assume that $\mathscr{A}$ is a nonempty set, every member of which is a \statementpadding - \lean*{Bookshelf/Enderton/Set/Relation} + \lean*{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_34\_a} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_34\_b} \paragraph{(a)}% @@ -5041,13 +5041,13 @@ Assume that $\mathscr{A}$ is a nonempty set, every member of which is a \end{proof} \subsection{\verified{Exercise 3.35}}% -\label{sub:exercise-3.35} +\hyperlabel{sub:exercise-3.35} Show that for any $R$ and $x$, we have $[x]_R = \img{R}{\{x\}}$. \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_35} Let $R$ and $x$ be arbitrary sets. @@ -5062,20 +5062,20 @@ Show that for any $R$ and $x$, we have $[x]_R = \img{R}{\{x\}}$. \end{proof} \subsection{\verified{Exercise 3.36}}% -\label{sub:exercise-3.36} +\hyperlabel{sub:exercise-3.36} Assume that $f \colon A \rightarrow B$ and that $R$ is an equivalence relation on $B$. Define $Q$ to be the set \begin{equation} - \label{sub:exercise-3.36-eq1} + \hyperlabel{sub:exercise-3.36-eq1} \{\pair{x, y} \in A \times A \mid \pair{f(x), f(y)} \in R\}. \end{equation} Show that $Q$ is an equivalence relation on $A$. \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_36} We prove that (i) $Q$ is \nameref{ref:reflexive} on $A$, (ii) $Q$ is @@ -5112,12 +5112,12 @@ Show that $Q$ is an equivalence relation on $A$. \end{proof} \subsection{\verified{Exercise 3.37}}% -\label{sub:exercise-3.37} +\hyperlabel{sub:exercise-3.37} Assume that $\Pi$ is a partition of a set $A$. Define the relation $R_\Pi$ as follows: \begin{equation} - \label{sub:exercise-3.37-eq1} + \hyperlabel{sub:exercise-3.37-eq1} xR_{\Pi}y \iff (\exists B \in \Pi)(x \in B \land y \in B). \end{equation} Show that $R_\Pi$ is an equivalence relation on $A$. @@ -5126,7 +5126,7 @@ Show that $R_\Pi$ is an equivalence relation on $A$. \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_37} We prove that (i) $R_\Pi$ is \nameref{ref:reflexive} on $B$, (ii) $R_\Pi$ is @@ -5165,7 +5165,7 @@ Show that $R_\Pi$ is an equivalence relation on $A$. \end{proof} \subsection{\verified{Exercise 3.38}}% -\label{sub:exercise-3.38} +\hyperlabel{sub:exercise-3.38} \nameref{sub:theorem-3p} shows that $A / R$ is a partition of $A$ whenever $R$ is an equivalence relation on $A$. @@ -5174,12 +5174,12 @@ Show that if we start with the equivalence relation $R_\Pi$ of the preceding \begin{proof} - \lean{Bookshelf/Enderton/Set/Relation} + \lean{Bookshelf/Enderton/Set/Chapter\_3} {Enderton.Set.Chapter\_3.exercise\_3\_38} By definition, \begin{equation} - \label{sub:exercise-3.38-eq1} + \hyperlabel{sub:exercise-3.38-eq1} R_\Pi = \{ (x, y) \mid (\exists B \in \Pi)(x \in B \land y \in B) \}. \end{equation} We prove that $A / R_\Pi = \Pi$. @@ -5258,7 +5258,7 @@ Show that if we start with the equivalence relation $R_\Pi$ of the preceding \end{proof} \subsection{\sorry{Exercise 3.39}}% -\label{sub:exercise-3.39} +\hyperlabel{sub:exercise-3.39} Assume that we start with an equivalence relation $R$ on $A$ and define $\Pi$ to be the partition $A / R$. @@ -5271,7 +5271,7 @@ Show that $R_\Pi$, as defined in Exercise 37, is just $R$. \end{proof} \subsection{\pending{Exercise 3.40}}% -\label{sub:exercise-3.40} +\hyperlabel{sub:exercise-3.40} Define an equivalence relation $R$ on the set $P$ of positive integers by $$mRn \iff m \text{ and } n \text{ have the same number of prime factors}.$$ @@ -5295,7 +5295,7 @@ Is there a function $f \colon P / R \rightarrow P / R$ such that \end{proof} \subsection{\sorry{Exercise 3.41}}% -\label{sub:exercise-3.41} +\hyperlabel{sub:exercise-3.41} Let $\mathbb{R}$ be the set of real numbers and define the relation $Q$ on $\mathbb{R} \times \mathbb{R}$ by $\pair{u, v}Q\pair{x, y}$ iff @@ -5317,7 +5317,7 @@ Let $\mathbb{R}$ be the set of real numbers and define the relation $Q$ on \end{proof} \subsection{\sorry{Exercise 3.42}}% -\label{sub:exercise-3.42} +\hyperlabel{sub:exercise-3.42} State precisely the "analogous results" mentioned in Theorem 3Q. (This will require extending the concept of compatibility in a suitable way.) diff --git a/Common/Real/Sequence.tex b/Common/Real/Sequence.tex index eb456da..e8edd00 100644 --- a/Common/Real/Sequence.tex +++ b/Common/Real/Sequence.tex @@ -10,15 +10,15 @@ \tableofcontents \section{Summations}% -\label{sec:summations} +\hyperlabel{sec:summations} \subsection{\verified{Arithmetic Series}}% -\label{sub:sum-arithmetic-series} +\hyperlabel{sub:sum-arithmetic-series} Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$. Then for some $n \in \mathbb{N}$, \begin{equation} - \label{sub:sum-arithmetic-series-eq1} + \hyperlabel{sub:sum-arithmetic-series-eq1} \sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}. \end{equation} @@ -30,7 +30,7 @@ Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$. Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$. By definition, for all $k \in \mathbb{N}$, \begin{equation} - \label{sub:sum-arithmetic-series-eq2} + \hyperlabel{sub:sum-arithmetic-series-eq2} a_k = (a_0 + kd). \end{equation} Define predicate $P(n)$ as "identity \eqref{sub:sum-arithmetic-series-eq1} @@ -73,12 +73,12 @@ Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$. \end{proof} \subsection{\verified{Geometric Series}}% -\label{sub:sum-geometric-series} +\hyperlabel{sub:sum-geometric-series} Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$. Then for some $n \in \mathbb{N}$, \begin{equation} - \label{sub:sum-geometric-series-eq1} + \hyperlabel{sub:sum-geometric-series-eq1} \sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}. \end{equation} @@ -90,7 +90,7 @@ Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$. Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$. By definition, for all $k \in \mathbb{N}$, \begin{equation} - \label{sub:sum-geometric-series-eq2} + \hyperlabel{sub:sum-geometric-series-eq2} a_k = a_0r^k. \end{equation} Define predicate $P(n)$ as "identity \eqref{sub:sum-geometric-series-eq1} diff --git a/preamble.tex b/preamble.tex index 2017eee..7443eb7 100644 --- a/preamble.tex +++ b/preamble.tex @@ -38,6 +38,9 @@ \hypersetup{colorlinks=true, linkcolor=blue, urlcolor=blue} \newcommand{\textref}[1]{\text{\nameref{#1}}} +\newcommand{\hyperlabel}[1]{% + \label{#1}% + \hypertarget{#1}{}} \newcommand\@leanlink[4]{% \textcolor{blue}{$\pmb{\exists}\;{-}\;$}\href{#1/#2.html\##3}{#4}}