From 5233126f055981d1778478a4a04db5d3edad78df Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Thu, 18 May 2023 13:03:59 -0600 Subject: [PATCH] Rename glossary to reference. --- Bookshelf/Apostol.tex | 190 ++++++++++++++++++----------------- Bookshelf/Enderton_Logic.tex | 6 +- 2 files changed, 99 insertions(+), 97 deletions(-) diff --git a/Bookshelf/Apostol.tex b/Bookshelf/Apostol.tex index c4c4176..901b2f2 100644 --- a/Bookshelf/Apostol.tex +++ b/Bookshelf/Apostol.tex @@ -15,13 +15,15 @@ \tableofcontents \begingroup -\renewcommand\thechapter{G} +\renewcommand\thechapter{R} +\setcounter{chapter}{0} +\addtocounter{chapter}{-1} -\chapter{Glossary}% -\label{chap:glossary} +\chapter{Reference}% +\label{chap:reference} \section{\defined{Characteristic Function}}% -\label{def:characteristic-function} +\label{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 @@ -35,7 +37,7 @@ The \textbf{characteristic function} of $S$ is the function $\mathcal{X}_S$ such \end{definition} \section{\defined{Infimum}}% -\label{def:infimum} +\label{ref:infimum} A number $B$ is called an \textbf{infimum} of a nonempty set $S$ if $B$ has the following two properties: @@ -52,36 +54,36 @@ Such a number $B$ is also known as the \textbf{greatest lower bound}. \end{definition} \section{\partial{Integrable}}% -\label{def:integrable} +\label{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 - $I$ such that \eqref{def:integral-bounded-function-eq2} holds. + $I$ such that \eqref{ref:integral-bounded-function-eq2} holds. If $f$ is integrable on $[a, b]$, we say that the integral $\int_a^b f(x) \mathop{dx}$ \textbf{exists}. \section{\partial{Integral of a Bounded Function}}% -\label{def:integral-bounded-function} +\label{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{def:integral-bounded-function-eq1} + \label{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{def:integral-bounded-function-eq2} + \label{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 - \eqref{def:integral-bounded-function-eq1}, then this number $I$ is called + \eqref{ref:integral-bounded-function-eq1}, then this number $I$ is called the \textbf{integral of $f$ from $a$ to $b$}, and is denoted by the symbol $\int_a^b f(x) \mathop{dx}$ or by $\int_a^b f$. If $a < b$, we define $\int_b^a f(x) \mathop{dx} = -\int_a^b f(x) \mathop{dx}$, - provided $f$ is \nameref{def:integrable} on $[a, b]$. + provided $f$ is \nameref{ref:integrable} on $[a, b]$. We also define $\int_a^a f(x) \mathop{dx} = 0$. The function $f$ is called the \textbf{integrand}, the numbers $a$ and $b$ are @@ -89,10 +91,10 @@ The function $f$ is called the \textbf{integrand}, the numbers $a$ and $b$ are \textbf{interval of integration}. \section{\partial{Integral of a Step Function}}% -\label{def:integral-step-function} +\label{ref:integral-step-function} -Let $s$ be a \nameref{def:step-function} defined on $[a, b]$, and let - $P = \{x_0, x_1, \ldots, x_n\}$ be a \nameref{def:partition} of $[a, b]$ +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 $[a, b]$ such that $s$ is constant on the open subintervals of $P$. Denote by $s_k$ the constant value that $s$ takes in the $k$th open subinterval of $P$, so that @@ -105,17 +107,17 @@ 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{\partial{Lower Integral}}% -\label{def:lower-integral} +\label{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 - \nameref{def:step-function}s below $f$. + \nameref{ref:step-function}s below $f$. That is, let $$S = \left\{ \int_a^b s(x) \mathop{dx} : s \leq f \right\}.$$ The number $\sup{S}$ is called the \textbf{lower integral of $f$}. It is denoted as $\ubar{I}(f)$. \section{\partial{Monotonic}}% -\label{def:monotonic} +\label{ref:monotonic} A function $f$ is called \textbf{monotonic} on set $S$ if it is increasing on $S$ or if it is decreasing on $S$. @@ -125,11 +127,11 @@ $f$ is said to be \textbf{strictly monotonic} if it is strictly increasing on A function $f$ is said to be \textbf{piecewise monotonic} on an interval if its graph consists of a finite number of monotonic pieces. In other words, $f$ is piecewise monotonic on $[a, b]$ if there is a - \nameref{def:partition} of $[a, b]$ such that $f$ is monotonic on each of + \nameref{ref:partition} of $[a, b]$ such that $f$ is monotonic on each of the open subintervals of $P$. \section{\defined{Partition}}% -\label{def:partition} +\label{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 @@ -151,9 +153,9 @@ A collection of points satisfying \eqref{sec:partition-eq1} is called a \end{definition} \section{\partial{Refinement}}% -\label{def:refinement} +\label{ref:refinement} -Let $P$ be a \nameref{def:partition} of closed interval $[a, b]$. +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 subdivision points to those already in $P$. @@ -161,10 +163,10 @@ $P'$ 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{def:step-function} +\label{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{def:partition} + \textbf{step function} if there is a \nameref{ref:partition} $P = \{x_0, x_1, \ldots, x_n\}$ of $[a, b]$ such that $s$ is constant on each open subinterval of $P$. That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$ @@ -183,7 +185,7 @@ That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$ \end{definition} \section{\defined{Supremum}}% -\label{def:supremum} +\label{ref:supremum} A number $B$ is called a \textbf{supremum} of a nonempty set $S$ if $B$ has the following two properties: @@ -200,11 +202,11 @@ Such a number $B$ is also known as the \textbf{least upper bound}. \end{definition} \section{\partial{Upper Integral}}% -\label{def:upper-integral} +\label{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 - \nameref{def:step-function}s above $f$. + \nameref{ref:step-function}s above $f$. That is, let $$T = \left\{ \int_a^b t(x) \mathop{dx} : f \leq t \right\}.$$ The number $\inf{T}$ is called the \textbf{upper integral of $f$}. It is denoted as $\bar{I}(f)$. @@ -241,7 +243,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum; {Apostol.Chapter\_I\_03.is\_lub\_neg\_set\_iff\_is\_glb\_set\_neg} Suppose $L = \sup{S}$ and fix $x \in S$. - By definition of the \nameref{def:supremum}, $x \leq L$ and $L$ is the + By definition of the \nameref{ref:supremum}, $x \leq L$ and $L$ is the smallest value satisfying this inequality. Negating both sides of the inequality yields $-x \geq -L$. Furthermore, $-L$ must be the largest value satisfying this inequality. @@ -268,7 +270,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum; Let $S$ be a nonempty set bounded below by $x$. Then $-S$ is nonempty and bounded above by $x$. By the \nameref{sec:completeness-axiom}, there exists a - \nameref{def:supremum} $L$ of $-S$. + \nameref{ref:supremum} $L$ of $-S$. By \nameref{sec:lemma-1}, $L$ is a supremum of $-S$ if and only if $-L$ is an infimum of $S$. @@ -433,7 +435,7 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers. \lean{Bookshelf/Apostol/Chapter\_I\_03} {Apostol.Chapter\_I\_03.sup\_imp\_exists\_gt\_sup\_sub\_delta} - By definition of a \nameref{def:supremum}, $\sup{S}$ is the least upper + By definition of a \nameref{ref:supremum}, $\sup{S}$ is the least upper bound of $S$. For the sake of contradiction, suppose for all $x \in S$, $x \leq \sup{S} - h$. @@ -459,7 +461,7 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers. \lean{Bookshelf/Apostol/Chapter\_I\_03} {Apostol.Chapter\_I\_03.inf\_imp\_exists\_lt\_inf\_add\_delta} - By definition of an \nameref{def:infimum}, $\inf{S}$ is the greatest lower + By definition of an \nameref{ref:infimum}, $\inf{S}$ is the greatest lower bound of $S$. For the sake of contradiction, suppose for all $x \in S$, $x \geq \inf{S} + h$. @@ -504,7 +506,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set Let $x \in C$. By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such that $x = a' + b'$. - By definition of a \nameref{def:supremum}, $a' \leq \sup{A}$. + By definition of a \nameref{ref:supremum}, $a' \leq \sup{A}$. Likewise, $b' \leq \sup{B}$. Therefore $a' + b' \leq \sup{A} + \sup{B}$. Since $x = a' + b'$ was arbitrarily chosen, it follows $\sup{A} + \sup{B}$ @@ -575,7 +577,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set Let $x \in C$. By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such that $x = a' + b'$. - By definition of an \nameref{def:infimum}, $a' \geq \inf{A}$. + By definition of an \nameref{ref:infimum}, $a' \geq \inf{A}$. Likewise, $b' \geq \inf{B}$. Therefore $a' + b' \geq \inf{A} + \inf{B}$. Since $x = a' + b'$ was arbitrarily chosen, it follows $\inf{A} + \inf{B}$ @@ -1740,8 +1742,8 @@ Now apply Exercises 4(a) and (b) to the bracket on the right. \label{sub:exercise-1.11.8} Let $S$ be a set of points on the real line. -Let $\mathcal{X}_S$ denote the \nameref{def:characteristic-function} of $S$. -Let $f$ be a \nameref{def:step-function} which takes the constant value +Let $\mathcal{X}_S$ denote the \nameref{ref:characteristic-function} of $S$. +Let $f$ be a \nameref{ref:step-function} which takes the constant value $c_k$ on the $k$th open subinterval $I_k$ of some partition of an interval $[a, b]$. Prove that for each $x$ in the union $I_1 \cup I_2 \cup \cdots \cup I_n$ we have @@ -1754,7 +1756,7 @@ This property is described by saying that every step function is a linear Let $x \in I_1 \cup I_2 \cup \cdots \cup I_n$ and $N = \{1, \ldots, n\}$. Let $k \in N$ such that $x \in I_k$. Consider an arbitrary $j \in N - \{k\}$. - By definition of a nameref{def:partition}, $I_j \cap I_k = \emptyset$. + By definition of a nameref{ref:partition}, $I_j \cap I_k = \emptyset$. That is, $I_j$ and $I_k$ are disjoint for all $j \in N - \{k\}$. Therefore, by definition of the characteristic function, $\mathcal{X}_{I_k}(x) = 1$ and $\mathcal{X}_{I_j}(x) = 0$ for all @@ -1780,7 +1782,7 @@ This property is described by saying that every step function is a linear \begin{theorem}[1.2] - Let $s$ and $t$ be \nameref{def:step-function}s on closed interval + Let $s$ and $t$ be \nameref{ref:step-function}s on closed interval $[a, b]$. Then $$\int_a^b \left[ s(x) + t(x) \right] \mathop{dx} = @@ -1791,7 +1793,7 @@ This property is described by saying that every step function is a linear \begin{proof} Let $s$ and $t$ be step functions on closed interval $[a, b]$. - By definition of a step function, there exists a \nameref{def:partition} + By definition of a step function, there exists a \nameref{ref:partition} $P_s$ such that $s$ is constant on each open subinterval of $P_s$. Likewise, there exists a partition $P_t$ such that $t$ is constant on each open subinterval of $P_t$. @@ -1804,7 +1806,7 @@ This property is described by saying that every step function is a linear $P$. Let $t_k$ denote the constant value of $t$ on the $k$th open subinterval of $P$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_a^b \left[ s(x) + t(x) \right] \mathop{dx} & = \sum_{k=1}^n (s_k + t_k) \cdot (x_k - x_{k-1}) \\ @@ -1823,7 +1825,7 @@ This property is described by saying that every step function is a linear \begin{theorem}[1.3] - Let $s$ be a \nameref{def:step-function} on closed interval $[a, b]$. + Let $s$ be a \nameref{ref:step-function} on closed interval $[a, b]$. For every real number $c$, we have $$\int_a^b c \cdot s(x) \mathop{dx} = c\int_a^b s(x) \mathop{dx}.$$ @@ -1832,13 +1834,13 @@ This property is described by saying that every step function is a linear \begin{proof} Let $s$ be a step function on closed interval $[a, b]$. - By definition of a step function, there exists a \nameref{def:partition} + By definition of a step function, there exists a \nameref{ref:partition} $P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open subinterval of $P$. Let $s_k$ denote the constant value of $s$ on the $k$th open subinterval of $P$. Then $c \cdot s$ is a step function with step partition $P$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_a^b c \cdot s(x) \mathop{dx} & = \sum_{k=1}^n c \cdot s_k \cdot (x_k - x_{k-1}) \\ @@ -1854,7 +1856,7 @@ This property is described by saying that every step function is a linear \begin{theorem}[1.4] - Let $s$ and $t$ be \nameref{def:step-function}s on closed interval + Let $s$ and $t$ be \nameref{ref:step-function}s on closed interval $[a, b]$. For every real $c_1$ and $c_2$, we have $$\int_a^b \left[ c_1s(x) + c_2t(x) \right] \mathop{dx} = @@ -1884,7 +1886,7 @@ This property is described by saying that every step function is a linear \begin{theorem}[1.5] - Let $s$ and $t$ be \nameref{def:step-function}s on closed interval + Let $s$ and $t$ be \nameref{ref:step-function}s on closed interval $[a, b]$. If $s(x) < t(x)$ for every $x$ in $[a, b]$, then $$\int_a^b s(x) \mathop{dx} < \int_a^b t(x) \mathop{dx}.$$ @@ -1894,7 +1896,7 @@ This property is described by saying that every step function is a linear \begin{proof} Let $s$ and $t$ be step functions on closed interval $[a, b]$. - By definition of a step function, there exists a \nameref{def:partition} + By definition of a step function, there exists a \nameref{ref:partition} $P_s$ such that $s$ is constant on each open subinterval of $P_s$. Likewise, there exists a partition $P_t$ such that $t$ is constant on each open subinterval of $P_t$. @@ -1907,7 +1909,7 @@ This property is described by saying that every step function is a linear $P$. Let $t_k$ denote the constant value of $t$ on the $k$th open subinterval of $P$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_a^b s(x) \mathop{dx} & = \sum_{k=1}^n s_k \cdot (x_k - x_{k-1}) \\ @@ -1923,7 +1925,7 @@ This property is described by saying that every step function is a linear \begin{theorem}[1.6] - Let $a, b, c \in \mathbb{R}$ and $s$ a \nameref{def:step-function} on the + Let $a, b, c \in \mathbb{R}$ and $s$ a \nameref{ref:step-function} on the smallest closed interval containing them. Then $$\int_a^c s(x) \mathop{dx} + \int_c^b s(x) \mathop{dx} + @@ -1935,7 +1937,7 @@ This property is described by saying that every step function is a linear WLOG, suppose $a < c < b$ and $s$ be a step function on closed interval $[a, b]$. - By definition of a step function, there exists a \nameref{def:partition} + By definition of a step function, there exists a \nameref{ref:partition} $P$ such that $s$ is constant on each open subinterval of $P$. Let $Q = \{x_0, x_1, \ldots, x_n\}$ be a refinement of $P$ that includes $c$ @@ -1944,7 +1946,7 @@ This property is described by saying that every step function is a linear that $x_i = c$. Let $s_k$ denote the constant value of $s$ on the $k$th open subinterval of $Q$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_a^b s(x) \mathop{dx} & = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1}) \\ @@ -1979,7 +1981,7 @@ This property is described by saying that every step function is a linear \begin{proof} Let $s$ be a step function on closed interval $[a, b]$. - By definition of a step function, there exists a \nameref{def:partition} + By definition of a step function, there exists a \nameref{ref:partition} $P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open subinterval of $P$. Let $s_k$ denote the constant value of $s$ on the $k$th open subinterval of @@ -1992,7 +1994,7 @@ This property is described by saying that every step function is a linear Let $t_k$ denote the value of $t$ on the $k$th open subinterval of $Q$. By construction, $t_k = s_k$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_{a+c}^{b+c} s(x - c) \mathop{dx} & = \int_{a+c}^{b+c} t(x) \mathop{dx} \\ @@ -2020,7 +2022,7 @@ This property is described by saying that every step function is a linear \begin{proof} Let $s$ be a step function on closed interval $[a, b]$. - By definition of a step function, there exists a \nameref{def:partition} + By definition of a step function, there exists a \nameref{ref:partition} $P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open subinterval of $P$. Let $s_i$ denote the value of $s$ on the $i$th open subinterval of $P$. @@ -2034,7 +2036,7 @@ This property is described by saying that every step function is a linear Then $t(x) = s(x / k)$ is a step function on closed interval $[ka, kb]$ with partition $Q = \{kx_0, kx_1, \ldots, kx_n\}$. Furthermore $t_i = s_i$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_{ka}^{kb} s(x / k) \mathop{dx} & = \int_{ka}^{kb} t(x) \mathop{dx} \\ @@ -2050,7 +2052,7 @@ This property is described by saying that every step function is a linear Then $t(x) = s(x / k)$ is a step function on closed interval $[kb, ka]$ with partition $Q = \{kx_n, kx_{n-1}, \ldots, kx_0\}$. Furthermore $t_i = s_i$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_{ka}^{kb} s(x / k) \mathop{dx} & = -\int_{kb}^{ka} s(x / k) \mathop{dx} \\ @@ -2102,7 +2104,7 @@ $\int_{-1}^3 \floor{x} \mathop{dx}$. $P = \{-1, 0, 1, 2, 3\} = \{x_0, x_1, x_2, x_3, x_4\}$. Let $s_k$ denote the constant value $s$ takes on the $k$th open subinterval of $P$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_{-1}^3 \floor{x} \mathop{dx} & = \sum_{k=1}^4 s_k \cdot (x_k - x_{k-1}) \\ @@ -2129,7 +2131,7 @@ $\int_{-1}^3 \left(\floor{x} + \floor{x + \frac{1}{2}}\right) \mathop{dx}$. \end{align*} Let $s_k$ denote the constant value $s$ takes on the $k$th open subinterval of $P$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_{-1}^3 \floor{x} + \floor{x + \frac{1}{2}} & = \sum_{k=1}^8 s_k \cdot (x_k - x_{k-1}) \\ @@ -2166,13 +2168,13 @@ Show that Let $s(x) = \floor{x}$ and $t(x) = \floor{-x}$, both with domain $[a, b]$. Let $x_1$, $\ldots$, $x_{n-1}$ denote the integers found in interval $(a, b)$. Then $P = \{x_0, x_1, \ldots, x_n\}$, $x_0 = a$ and $x_n = b$, is a step - \nameref{def:partition} of both $s$ and $t$. + \nameref{ref:partition} of both $s$ and $t$. Let $s_k$ and $t_k$ denote the constant values $s$ and $t$ take on the $k$th open subinterval of $P$ respectively. By \nameref{ssub:exercise-1.11.4b}, $\floor{-x} = -\floor{x} - 1$ for all $x$ in every open subinterval of $P$. That is, $s_k = -t_k - 1$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_a^b \floor{x} \mathop{dx} + \int_a^b \floor{-x} \mathop{dx} & = \sum_{k=1}^n s_k (x_k - x_{k-1}) + @@ -2197,11 +2199,11 @@ Prove that $\int_0^2 \floor{t^2} \mathop{dt} = 5 - \sqrt{2} - \sqrt{3}$. \begin{proof} Let $s(t) = \floor{t^2}$ with domain $[0, 2]$. - Then $s$ is a \nameref{def:step-function} with partition + Then $s$ is a \nameref{ref:step-function} with partition $P = \{0, 1, \sqrt{2}, \sqrt{3}, 2\} = \{x_0, x_1, \ldots, x_4\}$. Let $s_k$ denote the constant value that $s$ takes in the $k$th open subinterval of $P$. - By the \nameref{def:integral-step-function}, + By the \nameref{ref:integral-step-function}, \begin{align*} \int_0^2 \floor{t^2} \mathop{dt} & = \sum_{k=1}^4 s_k \cdot (x_k - x_{k-1}) \\ @@ -2220,7 +2222,7 @@ Compute $\int_{-3}^3 \floor{t^2} \mathop{dt}$. \begin{proof} Let $s(t) = \floor{t^2}$ with domain $[0, 3]$. - Then $s$ is a \nameref{def:step-function} with \nameref{def:partition} + Then $s$ is a \nameref{ref:step-function} with \nameref{ref:partition} \begin{align*} P & = \{\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}, @@ -2229,7 +2231,7 @@ Compute $\int_{-3}^3 \floor{t^2} \mathop{dt}$. \end{align*} Let $s_k$ denote the constant value that $s$ takes in the $k$th open subinterval of $P$. - By the \nameref{def:integral-step-function}, + By the \nameref{ref:integral-step-function}, \begin{align} \int_0^3 \floor{t^2} \mathop{dt} & = \sum_{k=1}^9 s_k \cdot (x_k - x_{k-1}) @@ -2267,11 +2269,11 @@ Compute $\int_0^9 \floor{\sqrt{t}} \mathop{dt}$. \begin{proof} Let $s(t) = \floor{\sqrt{t}}$ with domain $[0, 9]$. - Then $s$ is a \nameref{def:step-function} with \nameref{def:partition} + Then $s$ is a \nameref{ref:step-function} with \nameref{ref:partition} $P = \{0, 1, 4, 9\} = \{x_0, x_1, x_2, x_3\}$. Let $s_k$ denote the constant value that $s$ takes in the $k$th open subinterval of $P$. - By the \nameref{def:integral-step-function}, + By the \nameref{ref:integral-step-function}, \begin{align*} \int_0^9 \floor{\sqrt{t}} \mathop{dt} & = \sum_{k=1}^3 s_k \cdot (x_k - x_{k-1}) \\ @@ -2300,11 +2302,11 @@ If $n$ is a positive integer, prove that Let $n = 1$. Define $s(t) = \floor{\sqrt{t}}$ with domain $[0, 1]$. - Then $s$ is a \nameref{def:step-function} with \nameref{def:partition} + Then $s$ is a \nameref{ref:step-function} with \nameref{ref:partition} $P = \{0, 1\} = \{x_0, x_1\}$. Let $s_k$ denote the constant value of $s$ on the $k$th open subinterval of $P$. - By definition of the \nameref{def:integral-step-function}, the left-hand + By definition of the \nameref{ref:integral-step-function}, the left-hand side of \eqref{sub:exercise-1.15.7b-eq1} evaluates to \begin{align*} \int_0^{n^2} \floor{\sqrt{t}} \mathop{dt} @@ -2320,7 +2322,7 @@ If $n$ is a positive integer, prove that Let $n > 0$ be a positive integer and suppose $P(n)$ is true. Define $s(t) = \floor{\sqrt{t}}$ with domain $[0, (n + 1)^2]$. - Then $s$ is a \nameref{def:step-function} with \nameref{def:partition} + Then $s$ is a \nameref{ref:step-function} with \nameref{ref:partition} \begin{align*} P & = \{0, 1, 4, \ldots, n^2, (n + 1)^2\} \\ @@ -2328,7 +2330,7 @@ If $n$ is a positive integer, prove that \end{align*} Let $s_k$ denote the constant value of $s$ on the $k$th open subinterval of $P$. - By definition of the \nameref{def:integral-step-function}, it follows + By definition of the \nameref{ref:integral-step-function}, it follows that \begin{align*} & \int_0^{(n + 1)^2} s(x) \mathop{dx} \\ @@ -2405,8 +2407,8 @@ $\int_a^b s + \int_b^c s = \int_a^c s$. WLOG, suppose $a < b < c$. Let $s$ be a step function defined on closed interval $[a, c]$. - By definition of a \nameref{def:step-function}, there exists a - \nameref{def:partition} such that $s$ is constant on each open + By definition of a \nameref{ref:step-function}, there exists a + \nameref{ref:partition} such that $s$ is constant on each open subinterval of $P$. Let $Q = \{x_0, x_1, \ldots, x_n\}$ be a refinement of $P$ that includes $b$ as a subdivision point. @@ -2439,7 +2441,7 @@ $\int_a^b (s + t) = \int_a^b s + \int_a^b t$. \vspace{6pt} Let $s$ and $t$ be step functions on closed interval $[a, b]$. - By definition of a step function, there exists a \nameref{def:partition} + By definition of a step function, there exists a \nameref{ref:partition} $P_s$ such that $s$ is constant on each open subinterval of $P_s$. Likewise, there exists a partition $P_t$ such that $t$ is constant on each open subinterval of $P_t$. @@ -2486,7 +2488,7 @@ $\int_a^b c \cdot s = c \int_a^b s$. \vspace{6pt} Let $s$ be a step function on closed interval $[a, b]$. - By definition of a step function, there exists a \nameref{def:partition} + By definition of a step function, there exists a \nameref{ref:partition} $P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open subinterval of $P$. Let $s_k$ denote the constant value of $s$ on the $k$th open subinterval of @@ -2518,8 +2520,8 @@ $\int_{a+c}^{b+c} s(x) \mathop{dx} = \int_a^b s(x + c) \mathop{dx}$. \vspace{6pt} Let $s$ be a step function on closed interval $[a + c, b + c]$. - By definition of a \nameref{def:step-function}, there exists a - \nameref{def:partition} $P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is + By definition of a \nameref{ref:step-function}, there exists a + \nameref{ref:partition} $P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open subinterval of $P$. Let $s_k$ denote the constant value of $s$ on the $k$th open subinterval of $P$. @@ -2556,8 +2558,8 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \vspace{6pt} Let $s$ and $t$ be step functions on closed interval $[a, b]$. - By definition of a \nameref{def:step-function}, there exists a - \nameref{def:partition} $P_s$ such that $s$ is constant on each open + By definition of a \nameref{ref:step-function}, there exists a + \nameref{ref:partition} $P_s$ such that $s$ is constant on each open subinterval of $P_s$. Likewise, there exists a partition $P_t$ such that $t$ is constant on each open subinterval of $P_t$. @@ -2596,8 +2598,8 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \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} - for all \nameref{def:step-function}s $s$ and $t$ with $s \leq f \leq t$. - The function $f$ is \nameref{def:integrable} on $[a, b]$ if and only if + for all \nameref{ref:step-function}s $s$ and $t$ with $s \leq f \leq t$. + The function $f$ is \nameref{ref:integrable} on $[a, b]$ if and only if its upper and lower integrals are equal, in which case we have $$\int_a^b f(x) \mathop{dx} = \ubar{I}(f) = \bar{I}(f).$$ @@ -2627,10 +2629,10 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. By construction, $s \leq t$ for every $s$ in $S$ and $t$ in $T$. Therefore \nameref{sec:theorem-i.34} tells us $S$ has a - \nameref{def:supremum}, $T$ has an \nameref{def:infimum}, and + \nameref{ref:supremum}, $T$ has an \nameref{ref:infimum}, and $\sup{S} \leq \inf{T}$. - By definition of the \nameref{def:lower-integral}, $\ubar{I}(f) = \sup{S}$. - By definition of the \nameref{def:upper-integral}, $\bar{I}(f) = \inf{S}$. + By definition of the \nameref{ref:lower-integral}, $\ubar{I}(f) = \sup{S}$. + By definition of the \nameref{ref:upper-integral}, $\bar{I}(f) = \inf{S}$. Thus \eqref{sub:theorem-1.9-eq1} holds. \paragraph{(ii)}% @@ -2639,7 +2641,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. there exists one and only one number $I$ such that $$\int_a^b s(x) \mathop{dx} \leq I \leq \int_a^b t(x) \mathop{dx}$$ for every pair of step functions $s$ and $t$ satisfying - \eqref{def:integral-bounded-function-eq1}. + \eqref{ref:integral-bounded-function-eq1}. By \eqref{sub:theorem-1.9-eq1} and the definition of the supremum/infimum, this holds if and only if $\ubar{I}(f) = \bar{I}(f)$, concluding the proof. @@ -2654,7 +2656,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \begin{theorem}[1.10] - Let $f$ be a nonnegative function, \nameref{def:integrable} on an interval + Let $f$ be a nonnegative function, \nameref{ref:integrable} on an interval $[a, b]$, and let $Q$ denote the ordinate set of $f$ over $[a, b]$. Then $Q$ is measurable and its area is equal to the integral $\int_a^b f(x) \mathop{dx}$. @@ -2663,11 +2665,11 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \begin{proof} - Let $f$ be a nonnegative function, \nameref{def:integrable} on $[a, b]$. + Let $f$ be a nonnegative function, \nameref{ref:integrable} on $[a, b]$. By definition of integrability, there exists one and only one number $I$ such that $$\int_a^b s(x) \mathop{dx} \leq I \leq \int_a^b t(x) \mathop{dx}$$ for every pair of step functions $s$ and $t$ satisfying - \eqref{def:integral-bounded-function-eq1}. + \eqref{ref:integral-bounded-function-eq1}. In other words, $I$ is the one and only number that satisfies $$a(S) \leq I \leq a(T)$$ for every pair of step regions $S \subseteq Q \subseteq T$. @@ -2706,7 +2708,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. such that $$\int_a^b s(x) \mathop{dx} \leq I \leq \int_a^b t(x) \mathop{dx}$$ for every pair of step functions $s$ and $t$ satisfying - \eqref{def:integral-bounded-function-eq1}. + \eqref{ref:integral-bounded-function-eq1}. In other words, $I$ is the one and only number that satisfies $$a(S) \leq I \leq a(T)$$ for every pair of step regions $S \subseteq Q' \subseteq T$. @@ -2738,8 +2740,8 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \begin{theorem}[1.12] - If $f$ is \nameref{def:monotonic} on a closed interval $[a, b]$, then $f$ - is \nameref{def:integrable} on $[a, b]$. + If $f$ is \nameref{ref:monotonic} on a closed interval $[a, b]$, then $f$ + is \nameref{ref:integrable} on $[a, b]$. \end{theorem} @@ -2749,8 +2751,8 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. That is to say, either $f$ is increasing on $[a, b]$ or $f$ is decreasing on $[a, b]$. Because $f$ is on a closed interval, it is bounded. - By \nameref{sub:theorem-1.9}, $f$ has a \nameref{def:lower-integral} - $\ubar{I}(f)$, $f$ has an \nameref{def:upper-integral} $\bar{I}(f)$, + By \nameref{sub:theorem-1.9}, $f$ has a \nameref{ref:lower-integral} + $\ubar{I}(f)$, $f$ has an \nameref{ref:upper-integral} $\bar{I}(f)$, and $f$ is integrable if and only if $\ubar{I}(f) = \bar{I}(f)$. Consider a partition $P = \{x_0, x_1, \ldots, x_n\}$ of $[a, b]$ in which @@ -2770,7 +2772,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \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} - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_a^b s(x) \mathop{dx} & = \sum_{k=1}^n f(x_{k-1})\left[\frac{b - a}{n}\right] \\ @@ -2809,7 +2811,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. \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} - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_a^b s(x) \mathop{dx} & = \sum_{k=1}^n f(x_k)\left[\frac{b - a}{n}\right] \\ @@ -2863,7 +2865,7 @@ 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 $P$. Let $t$ be the step function above $f$ with constant value $f(x_k)$ on every $k$th open subinterval of $P$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_a^b s(x) \mathop{dx} & = \sum_{k=1}^n f(x_{k-1})\left[\frac{b - a}{n}\right] \\ @@ -2935,7 +2937,7 @@ 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 $P$. Let $t$ be the step function above $f$ with constant value $f(x_{k-1})$ on every $k$th open subinterval of $P$. - By definition of the \nameref{def:integral-step-function}, + By definition of the \nameref{ref:integral-step-function}, \begin{align*} \int_a^b s(x) \mathop{dx} & = \sum_{k=1}^n f(x_k) \left[\frac{b - a}{n}\right] \\ diff --git a/Bookshelf/Enderton_Logic.tex b/Bookshelf/Enderton_Logic.tex index 37c3384..a0e74f0 100644 --- a/Bookshelf/Enderton_Logic.tex +++ b/Bookshelf/Enderton_Logic.tex @@ -10,10 +10,10 @@ \tableofcontents \begingroup -\renewcommand\thechapter{G} +\renewcommand\thechapter{R} -\chapter{Glossary}% -\label{chap:glossary} +\chapter{Reference}% +\label{chap:reference} \endgroup