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Change coloring to distinguish "in progress" and "unverified".

finite-set-exercises
Joshua Potter 2023-06-30 11:50:34 -06:00
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Bookshelf/Apostol.tex
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@ -36,6 +36,18 @@ The \textbf{characteristic function} of $S$ is the function $\mathcal{X}_S$ such
\end{definition} \end{definition}
\section{\defined{Completeness Axiom}}%
\label{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 $B = \sup{S}$.
\begin{axiom}
\lean*{Mathlib/Data/Real/Basic}{Real.exists\_isLUB}
\end{axiom}
\section{\defined{Infimum}}% \section{\defined{Infimum}}%
\label{ref:infimum} \label{ref:infimum}
@ -53,7 +65,7 @@ Such a number $B$ is also known as the \textbf{greatest lower bound}.
\end{definition} \end{definition}
\section{\partial{Integrable}}% \section{\pending{Integrable}}%
\label{ref:integrable} \label{ref:integrable}
Let $f$ be a function defined and bounded on $[a, b]$. Let $f$ be a function defined and bounded on $[a, b]$.
@ -62,7 +74,7 @@ $f$ is said to be \textbf{integrable} if there exists one and only one number
If $f$ is integrable on $[a, b]$, we say that the integral If $f$ is integrable on $[a, b]$, we say that the integral
$\int_a^b f(x) \mathop{dx}$ \textbf{exists}. $\int_a^b f(x) \mathop{dx}$ \textbf{exists}.
\section{\partial{Integral of a Bounded Function}}% \section{\pending{Integral of a Bounded Function}}%
\label{ref:integral-bounded-function} \label{ref:integral-bounded-function}
Let $f$ be a function defined and bounded on $[a, b]$. Let $f$ be a function defined and bounded on $[a, b]$.
@ -90,7 +102,7 @@ The function $f$ is called the \textbf{integrand}, the numbers $a$ and $b$ are
called the \textbf{limits of integration}, and the interval $[a, b]$ the called the \textbf{limits of integration}, and the interval $[a, b]$ the
\textbf{interval of integration}. \textbf{interval of integration}.
\section{\partial{Integral of a Step Function}}% \section{\pending{Integral of a Step Function}}%
\label{ref:integral-step-function} \label{ref:integral-step-function}
Let $s$ be a \nameref{ref:step-function} defined on $[a, b]$, and let Let $s$ be a \nameref{ref:step-function} defined on $[a, b]$, and let
@ -106,7 +118,7 @@ The \textbf{integral of $s$ from $a$ to $b$}, denoted by the symbol
If $a < b$, we define $\int_b^a s(x) \mathop{dx} = -\int_a^b s(x) \mathop{dx}$. If $a < b$, we define $\int_b^a s(x) \mathop{dx} = -\int_a^b s(x) \mathop{dx}$.
We also define $\int_a^a s(x) \mathop{dx} = 0$. We also define $\int_a^a s(x) \mathop{dx} = 0$.
\section{\partial{Lower Integral}}% \section{\pending{Lower Integral}}%
\label{ref:lower-integral} \label{ref:lower-integral}
Let $f$ be a function bounded on $[a, b]$ and $S$ denote the set of numbers Let $f$ be a function bounded on $[a, b]$ and $S$ denote the set of numbers
@ -116,7 +128,7 @@ 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 $f$}. The number $\sup{S}$ is called the \textbf{lower integral of $f$}.
It is denoted as $\ubar{I}(f)$. It is denoted as $\ubar{I}(f)$.
\section{\partial{Monotonic}}% \section{\pending{Monotonic}}%
\label{ref:monotonic} \label{ref:monotonic}
A function $f$ is called \textbf{monotonic} on set $S$ if it is increasing on A function $f$ is called \textbf{monotonic} on set $S$ if it is increasing on
@ -152,7 +164,7 @@ A collection of points satisfying \eqref{sec:partition-eq1} is called a
\end{definition} \end{definition}
\section{\partial{Refinement}}% \section{\pending{Refinement}}%
\label{ref:refinement} \label{ref:refinement}
Let $P$ be a \nameref{ref:partition} of closed interval $[a, b]$. Let $P$ be a \nameref{ref:partition} of closed interval $[a, b]$.
@ -199,7 +211,7 @@ Such a number $B$ is also known as the \textbf{least upper bound}.
\end{definition} \end{definition}
\section{\partial{Upper Integral}}% \section{\pending{Upper Integral}}%
\label{ref:upper-integral} \label{ref:upper-integral}
Let $f$ be a function bounded on $[a, b]$ and $T$ denote the set of numbers Let $f$ be a function bounded on $[a, b]$ and $T$ denote the set of numbers
@ -214,18 +226,6 @@ It is denoted as $\bar{I}(f)$.
\chapter{A Set of Axioms for the Real-Number System}% \chapter{A Set of Axioms for the Real-Number System}%
\label{chap:set-axioms-real-number-system} \label{chap:set-axioms-real-number-system}
\section{\defined{Completeness Axiom}}%
\label{sec: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 $B = \sup{S}$.
\begin{axiom}
\lean*{Mathlib/Data/Real/Basic}{Real.exists\_isLUB}
\end{axiom}
\section{\verified{Lemma 1}}% \section{\verified{Lemma 1}}%
\label{sec:lemma-1} \label{sec:lemma-1}
@ -267,7 +267,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
Let $S$ be a nonempty set bounded below by $x$. Let $S$ be a nonempty set bounded below by $x$.
Then $-S$ is nonempty and bounded above by $x$. Then $-S$ is nonempty and bounded above by $x$.
By the \nameref{sec:completeness-axiom}, there exists a By the \nameref{ref:completeness-axiom}, there exists a
\nameref{ref: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 By \nameref{sec:lemma-1}, $L$ is a supremum of $-S$ if and only if $-L$ is an
infimum of $S$. infimum of $S$.
@ -731,7 +731,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
\end{axiom} \end{axiom}
\subsection{\partial{Exhaustion Property}}% \subsection{\pending{Exhaustion Property}}%
\label{sub:area-exhaustion-property} \label{sub:area-exhaustion-property}
Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so
@ -754,12 +754,12 @@ If there is one and only one number $c$ which satisfies the inequalities
\section{Exercises 1.7}% \section{Exercises 1.7}%
\label{sec:exercises-1.7} \label{sec:exercises-1.7}
\subsection{\partial{Exercise 1.7.1}}% \subsection{\pending{Exercise 1.7.1}}%
\label{sub:exercise-1.7.1} \label{sub:exercise-1.7.1}
Prove that each of the following sets is measurable and has zero area: Prove that each of the following sets is measurable and has zero area:
\subsubsection{\partial{Exercise 1.7.1a}}% \subsubsection{\pending{Exercise 1.7.1a}}%
\label{ssub:exercise-1.7.1a} \label{ssub:exercise-1.7.1a}
A set consisting of a single point. A set consisting of a single point.
@ -775,7 +775,7 @@ A set consisting of a single point.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.7.1b}}% \subsubsection{\pending{Exercise 1.7.1b}}%
\label{ssub:exercise-1.7.1b} \label{ssub:exercise-1.7.1b}
A set consisting of a finite number of points in a plane. A set consisting of a finite number of points in a plane.
@ -834,7 +834,7 @@ A set consisting of a finite number of points in a plane.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.7.1c}}% \subsubsection{\pending{Exercise 1.7.1c}}%
\label{ssub:exercise-1.7.1c} \label{ssub:exercise-1.7.1c}
The union of a finite collection of line segments in a plane. The union of a finite collection of line segments in a plane.
@ -897,7 +897,7 @@ The union of a finite collection of line segments in a plane.
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.7.2}}% \subsection{\pending{Exercise 1.7.2}}%
\label{sub:exercise-1.7.2} \label{sub:exercise-1.7.2}
Every right triangular region is measurable because it can be obtained as the Every right triangular region is measurable because it can be obtained as the
@ -948,7 +948,7 @@ Prove that every triangular region is measurable and that its area is one half
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.7.3}}% \subsection{\pending{Exercise 1.7.3}}%
\label{sub:exercise-1.7.3} \label{sub:exercise-1.7.3}
Prove that every trapezoid and every parallelogram is measurable and derive the Prove that every trapezoid and every parallelogram is measurable and derive the
@ -1053,14 +1053,14 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.7.4}}% \subsection{\pending{Exercise 1.7.4}}%
\label{sub:exercise-1.7.4} \label{sub:exercise-1.7.4}
Let $P$ be a polygon whose vertices are lattice points. Let $P$ be a polygon whose vertices are lattice points.
The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of
lattice points inside the polygon and $B$ denotes the number on the boundary. lattice points inside the polygon and $B$ denotes the number on the boundary.
\subsubsection{\partial{Exercise 1.7.4a}}% \subsubsection{\pending{Exercise 1.7.4a}}%
\label{ssub:exercise-1.7.4a} \label{ssub:exercise-1.7.4a}
Prove that the formula is valid for rectangles with sides parallel to the Prove that the formula is valid for rectangles with sides parallel to the
@ -1088,7 +1088,7 @@ Prove that the formula is valid for rectangles with sides parallel to the
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.7.4b}}% \subsubsection{\pending{Exercise 1.7.4b}}%
\label{ssub:exercise-1.7.4b} \label{ssub:exercise-1.7.4b}
Prove that the formula is valid for right triangles and parallelograms. Prove that the formula is valid for right triangles and parallelograms.
@ -1137,7 +1137,7 @@ Prove that the formula is valid for right triangles and parallelograms.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.7.4c}}% \subsubsection{\pending{Exercise 1.7.4c}}%
\label{ssub:exercise-1.7.4c} \label{ssub:exercise-1.7.4c}
Use induction on the number of edges to construct a proof for general polygons. Use induction on the number of edges to construct a proof for general polygons.
@ -1203,7 +1203,7 @@ Use induction on the number of edges to construct a proof for general polygons.
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.7.5}}% \subsection{\pending{Exercise 1.7.5}}%
\label{sub:exercise-1.7.5} \label{sub:exercise-1.7.5}
Prove that a triangle whose vertices are lattice points cannot be equilateral. Prove that a triangle whose vertices are lattice points cannot be equilateral.
@ -1239,7 +1239,7 @@ ways, using Exercises 2 and 4.]
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.7.6}}% \subsection{\pending{Exercise 1.7.6}}%
\label{sub:exercise-1.7.6} \label{sub:exercise-1.7.6}
Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all
@ -1324,7 +1324,7 @@ Prove that the set function $n$ satisfies the first three axioms for area.
\section{Exercises 1.11}% \section{Exercises 1.11}%
\label{sec:exercises-1-11} \label{sec:exercises-1-11}
\subsection{\partial{Exercise 1.11.4}}% \subsection{\pending{Exercise 1.11.4}}%
\label{sub:exercise-1.11.4} \label{sub:exercise-1.11.4}
Prove that the greatest-integer function has the properties indicated: Prove that the greatest-integer function has the properties indicated:
@ -1442,7 +1442,7 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.11.4d}}% \subsubsection{\pending{Exercise 1.11.4d}}%
\label{ssub:exercise-1.11.4d} \label{ssub:exercise-1.11.4d}
$\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
@ -1456,7 +1456,7 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.11.4e}}% \subsubsection{\pending{Exercise 1.11.4e}}%
\label{ssub:exercise-1.11.4e} \label{ssub:exercise-1.11.4e}
$\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
@ -1470,7 +1470,7 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
\end{proof} \end{proof}
\subsection{\partial{Hermite's Identity}}% \subsection{\pending{Hermite's Identity}}%
\label{sub:hermites-identity} \label{sub:hermites-identity}
\label{sub:exercise-1.11.5} \label{sub:exercise-1.11.5}
@ -1559,7 +1559,7 @@ State and prove such a generalization.
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.11.6}}% \subsection{\pending{Exercise 1.11.6}}%
\label{sub:exercise-1.11.6} \label{sub:exercise-1.11.6}
Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
@ -1602,7 +1602,7 @@ Prove that the number of lattice points in $S$ is equal to the sum
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.11.7}}% \subsection{\pending{Exercise 1.11.7}}%
\label{sub:exercise-1.11.7} \label{sub:exercise-1.11.7}
If $a$ and $b$ are positive integers with no common factor, we have the formula If $a$ and $b$ are positive integers with no common factor, we have the formula
@ -1612,7 +1612,7 @@ When $b = 1$, the sum on the left is understood to be $0$.
\note{When $b = 1$, the proofs of (a) and (b) are trivial. We continue under the \note{When $b = 1$, the proofs of (a) and (b) are trivial. We continue under the
assumption $b > 1$.} assumption $b > 1$.}
\subsubsection{\partial{Exercise 1.11.7a}}% \subsubsection{\pending{Exercise 1.11.7a}}%
\label{ssub:exercise-1.11.7a} \label{ssub:exercise-1.11.7a}
Derive this result by a geometric argument, counting lattice points in a right Derive this result by a geometric argument, counting lattice points in a right
@ -1694,7 +1694,7 @@ Derive this result by a geometric argument, counting lattice points in a right
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.11.7b}}% \subsubsection{\pending{Exercise 1.11.7b}}%
\label{ssub:exercise-1.11.7b} \label{ssub:exercise-1.11.7b}
Derive the result analytically as follows: Derive the result analytically as follows:
@ -1736,7 +1736,7 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.11.8}}% \subsection{\pending{Exercise 1.11.8}}%
\label{sub:exercise-1.11.8} \label{sub:exercise-1.11.8}
Let $S$ be a set of points on the real line. Let $S$ be a set of points on the real line.
@ -1774,7 +1774,7 @@ This property is described by saying that every step function is a linear
\section{Properties of the Integral of a Step Function}% \section{Properties of the Integral of a Step Function}%
\label{sec:properties-integral-step-function} \label{sec:properties-integral-step-function}
\subsection{\partial{Additive Property}}% \subsection{\pending{Additive Property}}%
\label{sub:step-additive-property} \label{sub:step-additive-property}
\label{sub:theorem-1.2} \label{sub:theorem-1.2}
@ -1817,7 +1817,7 @@ This property is described by saying that every step function is a linear
\end{proof} \end{proof}
\subsection{\partial{Homogeneous Property}}% \subsection{\pending{Homogeneous Property}}%
\label{sub:step-homogeneous-property} \label{sub:step-homogeneous-property}
\label{sub:theorem-1.3} \label{sub:theorem-1.3}
@ -1848,7 +1848,7 @@ This property is described by saying that every step function is a linear
\end{proof} \end{proof}
\subsection{\partial{Linearity Property}}% \subsection{\pending{Linearity Property}}%
\label{sub:step-linearity-property} \label{sub:step-linearity-property}
\label{sub:theorem-1.4} \label{sub:theorem-1.4}
@ -1878,7 +1878,7 @@ This property is described by saying that every step function is a linear
\end{proof} \end{proof}
\subsection{\partial{Comparison Theorem}}% \subsection{\pending{Comparison Theorem}}%
\label{sub:step-comparison-theorem} \label{sub:step-comparison-theorem}
\label{sub:theorem-1.5} \label{sub:theorem-1.5}
@ -1917,7 +1917,7 @@ This property is described by saying that every step function is a linear
\end{proof} \end{proof}
\subsection{\partial{Additivity With Respect to the Interval of Integration}}% \subsection{\pending{Additivity With Respect to the Interval of Integration}}%
\label{sub:step-additivity-with-respect-interval-integration} \label{sub:step-additivity-with-respect-interval-integration}
\label{sub:theorem-1.6} \label{sub:theorem-1.6}
@ -1963,7 +1963,7 @@ This property is described by saying that every step function is a linear
\end{proof} \end{proof}
\subsection{\partial{Invariance Under Translation}}% \subsection{\pending{Invariance Under Translation}}%
\label{sub:step-invariance-under-translation} \label{sub:step-invariance-under-translation}
\label{sub:theorem-1.7} \label{sub:theorem-1.7}
@ -2004,7 +2004,7 @@ This property is described by saying that every step function is a linear
\end{proof} \end{proof}
\subsection{\partial{Expansion or Contraction of the Interval of Integration}}% \subsection{\pending{Expansion or Contraction of the Interval of Integration}}%
\label{sub:step-expansion-contraction-interval-integration} \label{sub:step-expansion-contraction-interval-integration}
\label{sub:theorem-1.8} \label{sub:theorem-1.8}
@ -2064,7 +2064,7 @@ This property is described by saying that every step function is a linear
\end{proof} \end{proof}
\subsection{\partial{Reflection Property}}% \subsection{\pending{Reflection Property}}%
\label{sub:step-reflection-property} \label{sub:step-reflection-property}
Let $s$ be a step function on closed interval $[a, b]$. Let $s$ be a step function on closed interval $[a, b]$.
@ -2085,12 +2085,12 @@ Then
\section{Exercises 1.15}% \section{Exercises 1.15}%
\label{sec:exercises-1.15} \label{sec:exercises-1.15}
\subsection{\partial{Exercise 1.15.1}}% \subsection{\pending{Exercise 1.15.1}}%
\label{sub:exercise-1.15.1} \label{sub:exercise-1.15.1}
Compute the value of each of the following integrals. Compute the value of each of the following integrals.
\subsubsection{\partial{Exercise 1.15.1a}}% \subsubsection{\pending{Exercise 1.15.1a}}%
\label{ssub:exercise-1.15.1a} \label{ssub:exercise-1.15.1a}
$\int_{-1}^3 \floor{x} \mathop{dx}$. $\int_{-1}^3 \floor{x} \mathop{dx}$.
@ -2112,7 +2112,7 @@ $\int_{-1}^3 \floor{x} \mathop{dx}$.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.15.1c}}% \subsubsection{\pending{Exercise 1.15.1c}}%
\label{ssub:exercise-1.15.1c} \label{ssub:exercise-1.15.1c}
$\int_{-1}^3 \left(\floor{x} + \floor{x + \frac{1}{2}}\right) \mathop{dx}$. $\int_{-1}^3 \left(\floor{x} + \floor{x + \frac{1}{2}}\right) \mathop{dx}$.
@ -2140,7 +2140,7 @@ $\int_{-1}^3 \left(\floor{x} + \floor{x + \frac{1}{2}}\right) \mathop{dx}$.
\end{proof} \end{proof}
\subsubsection{\partial{Exericse 1.15.1e}}% \subsubsection{\pending{Exericse 1.15.1e}}%
\label{ssub:exercise-1.15.1e} \label{ssub:exercise-1.15.1e}
$\int_{-1}^3 \floor{2x} \mathop{dx}$. $\int_{-1}^3 \floor{2x} \mathop{dx}$.
@ -2155,7 +2155,7 @@ $\int_{-1}^3 \floor{2x} \mathop{dx}$.
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.15.3}}% \subsection{\pending{Exercise 1.15.3}}%
\label{sub:exercise-1.15.3} \label{sub:exercise-1.15.3}
Show that Show that
@ -2186,10 +2186,10 @@ Show that
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.15.5}}% \subsection{\pending{Exercise 1.15.5}}%
\label{sub:exercise-1.15.5} \label{sub:exercise-1.15.5}
\subsubsection{\partial{Exercise 1.15.5a}}% \subsubsection{\pending{Exercise 1.15.5a}}%
\label{ssub:exercise-1.15.5a} \label{ssub:exercise-1.15.5a}
Prove that $\int_0^2 \floor{t^2} \mathop{dt} = 5 - \sqrt{2} - \sqrt{3}$. Prove that $\int_0^2 \floor{t^2} \mathop{dt} = 5 - \sqrt{2} - \sqrt{3}$.
@ -2212,7 +2212,7 @@ Prove that $\int_0^2 \floor{t^2} \mathop{dt} = 5 - \sqrt{2} - \sqrt{3}$.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.15.5b}}% \subsubsection{\pending{Exercise 1.15.5b}}%
\label{ssub:exercise-1.15.5b} \label{ssub:exercise-1.15.5b}
Compute $\int_{-3}^3 \floor{t^2} \mathop{dt}$. Compute $\int_{-3}^3 \floor{t^2} \mathop{dt}$.
@ -2256,10 +2256,10 @@ Compute $\int_{-3}^3 \floor{t^2} \mathop{dt}$.
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.15.7}}% \subsection{\pending{Exercise 1.15.7}}%
\label{sub:exercise-1.15.7} \label{sub:exercise-1.15.7}
\subsubsection{\partial{Exercise 1.15.7a}}% \subsubsection{\pending{Exercise 1.15.7a}}%
\label{ssub:exercise-1.15.7a} \label{ssub:exercise-1.15.7a}
Compute $\int_0^9 \floor{\sqrt{t}} \mathop{dt}$. Compute $\int_0^9 \floor{\sqrt{t}} \mathop{dt}$.
@ -2281,7 +2281,7 @@ Compute $\int_0^9 \floor{\sqrt{t}} \mathop{dt}$.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.15.7b}}% \subsubsection{\pending{Exercise 1.15.7b}}%
\label{ssub:exercise-1.15.7b} \label{ssub:exercise-1.15.7b}
If $n$ is a positive integer, prove that If $n$ is a positive integer, prove that
@ -2356,7 +2356,7 @@ If $n$ is a positive integer, prove that
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.15.9}}% \subsection{\pending{Exercise 1.15.9}}%
\label{sub:exercise-1.15.9} \label{sub:exercise-1.15.9}
Show that the following property is equivalent to Show that the following property is equivalent to
@ -2378,7 +2378,7 @@ Show that the following property is equivalent to
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.15.11}}% \subsection{\pending{Exercise 1.15.11}}%
\label{sub:exercise-1.15.11} \label{sub:exercise-1.15.11}
If we instead defined the integral of step functions as If we instead defined the integral of step functions as
@ -2389,7 +2389,7 @@ If we instead defined the integral of step functions as
a new and different theory of integration would result. a new and different theory of integration would result.
Which of the following properties would remain valid in this new theory? Which of the following properties would remain valid in this new theory?
\subsubsection{\partial{Exercise 1.15.11a}}% \subsubsection{\pending{Exercise 1.15.11a}}%
\label{ssub:exercise-1.15.11a} \label{ssub:exercise-1.15.11a}
$\int_a^b s + \int_b^c s = \int_a^c s$. $\int_a^b s + \int_b^c s = \int_a^c s$.
@ -2425,7 +2425,7 @@ $\int_a^b s + \int_b^c s = \int_a^c s$.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.15.11b}}% \subsubsection{\pending{Exercise 1.15.11b}}%
\label{ssub:exercise-1.15.11b} \label{ssub:exercise-1.15.11b}
$\int_a^b (s + t) = \int_a^b s + \int_a^b t$. $\int_a^b (s + t) = \int_a^b s + \int_a^b t$.
@ -2472,7 +2472,7 @@ $\int_a^b (s + t) = \int_a^b s + \int_a^b t$.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.15.11c}}% \subsubsection{\pending{Exercise 1.15.11c}}%
\label{ssub:exercise-1.15.11c} \label{ssub:exercise-1.15.11c}
$\int_a^b c \cdot s = c \int_a^b s$. $\int_a^b c \cdot s = c \int_a^b s$.
@ -2504,7 +2504,7 @@ $\int_a^b c \cdot s = c \int_a^b s$.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.15.11d}}% \subsubsection{\pending{Exercise 1.15.11d}}%
\label{ssub:exercise-1.15.11d} \label{ssub:exercise-1.15.11d}
$\int_{a+c}^{b+c} s(x) \mathop{dx} = \int_a^b s(x + c) \mathop{dx}$. $\int_{a+c}^{b+c} s(x) \mathop{dx} = \int_a^b s(x + c) \mathop{dx}$.
@ -2542,7 +2542,7 @@ $\int_{a+c}^{b+c} s(x) \mathop{dx} = \int_a^b s(x + c) \mathop{dx}$.
\end{proof} \end{proof}
\subsubsection{\partial{Exercise 1.15.11e}}% \subsubsection{\pending{Exercise 1.15.11e}}%
\label{ssub:exercise-1.15.11e} \label{ssub:exercise-1.15.11e}
If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$. If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
@ -2583,7 +2583,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\section{Upper and Lower Integrals}% \section{Upper and Lower Integrals}%
\label{sec:upper-lower-integrals} \label{sec:upper-lower-integrals}
\subsection{\partial{Theorem 1.9}}% \subsection{\pending{Theorem 1.9}}%
\label{sub:theorem-1.9} \label{sub:theorem-1.9}
\begin{theorem}[1.9] \begin{theorem}[1.9]
@ -2649,7 +2649,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\section{The Area of an Ordinate Set Expressed as an Integral}% \section{The Area of an Ordinate Set Expressed as an Integral}%
\label{sec:area-ordinate-set-expressed-integral} \label{sec:area-ordinate-set-expressed-integral}
\subsection{\partial{Theorem 1.10}}% \subsection{\pending{Theorem 1.10}}%
\label{sub:theorem-1.10} \label{sub:theorem-1.10}
\begin{theorem}[1.10] \begin{theorem}[1.10]
@ -2676,7 +2676,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\end{proof} \end{proof}
\subsection{\partial{Theorem 1.11}}% \subsection{\pending{Theorem 1.11}}%
\label{sub:theorem-1.11} \label{sub:theorem-1.11}
\begin{theorem}[1.11] \begin{theorem}[1.11]
@ -2733,7 +2733,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
{Integrability of Bounded Monotonic \texorpdfstring{\\}{}Functions} {Integrability of Bounded Monotonic \texorpdfstring{\\}{}Functions}
\label{sec:integrability-bounded-monotonic-functions} \label{sec:integrability-bounded-monotonic-functions}
\subsection{\partial{Theorem 1.12}}% \subsection{\pending{Theorem 1.12}}%
\label{sub:theorem-1.12} \label{sub:theorem-1.12}
\begin{theorem}[1.12] \begin{theorem}[1.12]
@ -2837,7 +2837,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\end{proof} \end{proof}
\subsection{\partial{Theorem 1.13}}% \subsection{\pending{Theorem 1.13}}%
\label{sub:theorem-1.13} \label{sub:theorem-1.13}
\begin{theorem}[1.13] \begin{theorem}[1.13]
@ -2909,7 +2909,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\end{proof} \end{proof}
\subsection{\partial{Theorem 1.14}}% \subsection{\pending{Theorem 1.14}}%
\label{sub:theorem-1.14} \label{sub:theorem-1.14}
\begin{theorem}[1.14] \begin{theorem}[1.14]
@ -2981,7 +2981,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\end{proof} \end{proof}
\subsection{\unverified{% \subsection{\sorry{%
Integral of \texorpdfstring{$\int_0^b x^p \mathop{dx}$}{int-x-p} when Integral of \texorpdfstring{$\int_0^b x^p \mathop{dx}$}{int-x-p} when
\texorpdfstring{$p$}{p} is a Positive Integer}}% \texorpdfstring{$p$}{p} is a Positive Integer}}%
\label{sub:calculation-integral-int-x-p-p-positive-integer} \label{sub:calculation-integral-int-x-p-p-positive-integer}

2
Bookshelf/Enderton/Logic.tex
View File

@ -23,7 +23,7 @@
\chapter{Useful Facts About Sets}% \chapter{Useful Facts About Sets}%
\label{chap:useful-facts-about-sets} \label{chap:useful-facts-about-sets}
\section{\unverified{Lemma 0A}}% \section{\sorry{Lemma 0A}}%
\label{sec:lemma-0a} \label{sec:lemma-0a}
Assume that $\langle x_1, \ldots, x_m \rangle = Assume that $\langle x_1, \ldots, x_m \rangle =

90
Bookshelf/Enderton/Set.tex
View File

@ -25,13 +25,13 @@
\chapter{Reference}% \chapter{Reference}%
\label{chap:reference} \label{chap:reference}
\section{\partial{Axiom of Choice, First Form}}% \section{\pending{Axiom of Choice, First Form}}%
\label{ref:axiom-of-choice-1} \label{ref:axiom-of-choice-1}
For any relation $R$ there is a function $H \subseteq R$ with For any relation $R$ there is a function $H \subseteq R$ with
$\dom{H} = \dom{R}$. $\dom{H} = \dom{R}$.
\section{\partial{Axiom of Choice, Second Form}}% \section{\pending{Axiom of Choice, Second Form}}%
\label{ref:axiom-of-choice-2} \label{ref:axiom-of-choice-2}
For any set $I$ and any function $H$ with domain $I$, if $H(i) \neq \emptyset$ For any set $I$ and any function $H$ with domain $I$, if $H(i) \neq \emptyset$
@ -73,7 +73,7 @@ There is a set having no members:
\end{axiom} \end{axiom}
\section{\partial{Equivalence Relation}}% \section{\pending{Equivalence Relation}}%
\label{ref:equivalence-relation} \label{ref:equivalence-relation}
Relation $R$ is an \textbf{equivalence relation} if and only if $R$ is a binary Relation $R$ is an \textbf{equivalence relation} if and only if $R$ is a binary
@ -242,7 +242,7 @@ Given \nameref{ref:relation} $R$, the \textbf{range} of $R$, denoted $\ran{R}$,
\end{definition} \end{definition}
\section{\partial{Reflexive Relation}}% \section{\pending{Reflexive Relation}}%
\label{ref:reflexive-relation} \label{ref:reflexive-relation}
A binary relation $R$ is \textbf{reflexive} on $A$ if and only if $xRx$ for all A binary relation $R$ is \textbf{reflexive} on $A$ if and only if $xRx$ for all
@ -284,7 +284,7 @@ For each formula $\phi$ not containing $B$, the following is an axiom:
\end{axiom} \end{axiom}
\section{\partial{Symmetric Relation}}% \section{\pending{Symmetric Relation}}%
\label{ref:symmetric-relation} \label{ref:symmetric-relation}
A binary relation $R$ is \textbf{symmetric} on $A$ if and only if whenever A binary relation $R$ is \textbf{symmetric} on $A$ if and only if whenever
@ -302,7 +302,7 @@ The \textbf{symmetric difference} $A + B$ of sets $A$ and $B$ is the set
\end{definition} \end{definition}
\section{\partial{Transitive Relation}}% \section{\pending{Transitive Relation}}%
\label{ref:transitive-relation} \label{ref:transitive-relation}
A binary relation $R$ is \textbf{transitive} on $A$ if and only if whenever A binary relation $R$ is \textbf{transitive} on $A$ if and only if whenever
@ -505,7 +505,7 @@ Show that $\{\{x\}, \{x, y\}\} \in \powerset{\powerset{B}}.$
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.5}}% \subsection{\unverified{Exercise 1.5}}%
\label{sub:exercise-1.5} \label{sub:exercise-1.5}
Define the rank of a set $c$ to be the least $\alpha$ such that Define the rank of a set $c$ to be the least $\alpha$ such that
@ -545,7 +545,7 @@ Compute the rank of
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.6}}% \subsection{\unverified{Exercise 1.6}}%
\label{sub:exercise-1.6} \label{sub:exercise-1.6}
We have stated that $V_{\alpha + 1} = A \cup \powerset{V_\alpha}$. We have stated that $V_{\alpha + 1} = A \cup \powerset{V_\alpha}$.
@ -593,7 +593,7 @@ Prove this at least for $\alpha < 3$.
\end{proof} \end{proof}
\subsection{\partial{Exercise 1.7}}% \subsection{\unverified{Exercise 1.7}}%
\label{sub:exercise-1.7} \label{sub:exercise-1.7}
List all the members of $V_3$. List all the members of $V_3$.
@ -641,7 +641,7 @@ List all the members of $V_4$.
\section{Axioms}% \section{Axioms}%
\label{sec:axioms} \label{sec:axioms}
\subsection{\partial{Theorem 2A}}% \subsection{\unverified{Theorem 2A}}%
\label{sub:theorem-2a} \label{sub:theorem-2a}
\begin{theorem}[2A] \begin{theorem}[2A]
@ -665,7 +665,7 @@ List all the members of $V_4$.
\end{proof} \end{proof}
\subsection{\partial{Theorem 2B}}% \subsection{\unverified{Theorem 2B}}%
\label{sub:theorem-2b} \label{sub:theorem-2b}
\begin{theorem}[2B] \begin{theorem}[2B]
@ -1103,7 +1103,7 @@ For any sets $A$, $B$, and $C$,
\end{proof} \end{proof}
\subsection{\partial{General Distributive Laws}}% \subsection{\unverified{General Distributive Laws}}%
\label{sub:general-distributive-laws} \label{sub:general-distributive-laws}
For any sets $A$ and $\mathscr{B}$, For any sets $A$ and $\mathscr{B}$,
@ -1163,7 +1163,7 @@ For any sets $A$ and $\mathscr{B}$,
\end{proof} \end{proof}
\subsection{\partial{General De Morgan's Laws}}% \subsection{\unverified{General De Morgan's Laws}}%
\label{sub:general-de-morgans-laws} \label{sub:general-de-morgans-laws}
For any set $C$ and $\mathscr{A} \neq \emptyset$, For any set $C$ and $\mathscr{A} \neq \emptyset$,
@ -1590,7 +1590,7 @@ Under what conditions does equality hold?
\end{proof} \end{proof}
\subsection{\partial{Exercise 2.8}}% \subsection{\unverified{Exercise 2.8}}%
\label{sub:exercise-2.8} \label{sub:exercise-2.8}
Show that there is no set to which every singleton (that is, every set of the Show that there is no set to which every singleton (that is, every set of the
@ -1990,7 +1990,7 @@ Show that the following four conditions are equivalent.
\end{proof} \end{proof}
\subsection{\partial{Exercise 2.18}}% \subsection{\unverified{Exercise 2.18}}%
\label{sub:exercise-2.18} \label{sub:exercise-2.18}
Assume that $A$ and $B$ are subsets of $S$. Assume that $A$ and $B$ are subsets of $S$.
@ -2195,7 +2195,7 @@ Show that if $A$ and $B$ are nonempty sets, then
\end{proof} \end{proof}
\subsection{\partial{Exercise 2.23}}% \subsection{\unverified{Exercise 2.23}}%
\label{sub:exercise-2.23} \label{sub:exercise-2.23}
Show that if $\mathscr{B}$ is nonempty, then Show that if $\mathscr{B}$ is nonempty, then
@ -2856,7 +2856,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
\end{proof} \end{proof}
\subsection{\partial{Theorem 3J}}% \subsection{\pending{Theorem 3J}}%
\label{sub:theorem-3j} \label{sub:theorem-3j}
\begin{theorem}[3J] \begin{theorem}[3J]
@ -2942,7 +2942,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
\end{proof} \end{proof}
\subsection{\partial{Theorem 3K(a)}}% \subsection{\pending{Theorem 3K(a)}}%
\label{sub:theorem-3k-a} \label{sub:theorem-3k-a}
\begin{theorem}[3K(a)] \begin{theorem}[3K(a)]
@ -3010,7 +3010,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
\end{proof} \end{proof}
\subsection{\partial{Theorem 3K(b)}}% \subsection{\pending{Theorem 3K(b)}}%
\label{sub:theorem-3k-b} \label{sub:theorem-3k-b}
\begin{theorem}[3K(b)] \begin{theorem}[3K(b)]
@ -3093,7 +3093,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
\end{proof} \end{proof}
\subsection{\partial{Theorem 3K(c)}}% \subsection{\pending{Theorem 3K(c)}}%
\label{sub:theorem-3k-c} \label{sub:theorem-3k-c}
\begin{theorem}[3K(c)] \begin{theorem}[3K(c)]
@ -3143,7 +3143,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
\end{proof} \end{proof}
\subsection{\partial{Corollary 3L}}% \subsection{\pending{Corollary 3L}}%
\label{sub:corollary-3l} \label{sub:corollary-3l}
\begin{theorem}[3L] \begin{theorem}[3L]
@ -3175,7 +3175,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
\section{Equivalence Relations}% \section{Equivalence Relations}%
\label{sec:equivalence-relations} \label{sec:equivalence-relations}
\subsection{\unverified{Theorem 3M}}% \subsection{\sorry{Theorem 3M}}%
\label{sub:theorem-3m} \label{sub:theorem-3m}
\begin{theorem}[3M] \begin{theorem}[3M]
@ -3327,7 +3327,7 @@ Show that $A \times \bigcup \mathscr{B} =
\end{proof} \end{proof}
\subsection{\partial{Exercise 3.4}}% \subsection{\unverified{Exercise 3.4}}%
\label{sub:exercise-3.4} \label{sub:exercise-3.4}
Show that there is no set to which every ordered pair belongs. Show that there is no set to which every ordered pair belongs.
@ -3656,7 +3656,7 @@ Discuss the result of replacing the union operation by the intersection
\end{answer} \end{answer}
\subsection{\partial{Exercise 3.10}}% \subsection{\unverified{Exercise 3.10}}%
\label{sub:exercise-3.10} \label{sub:exercise-3.10}
Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
@ -3681,7 +3681,7 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
\end{answer} \end{answer}
\subsection{\unverified{Exercise 3.11}}% \subsection{\sorry{Exercise 3.11}}%
\label{sub:exercise-3.11} \label{sub:exercise-3.11}
Prove the following version (for functions) of the extensionality principle: Prove the following version (for functions) of the extensionality principle:
@ -3695,7 +3695,7 @@ Then $F = G$.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.12}}% \subsection{\sorry{Exercise 3.12}}%
\label{sub:exercise-3.12} \label{sub:exercise-3.12}
Assume that $f$ and $g$ are functions and show that Assume that $f$ and $g$ are functions and show that
@ -3708,7 +3708,7 @@ Assume that $f$ and $g$ are functions and show that
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.13}}% \subsection{\sorry{Exercise 3.13}}%
\label{sub:exercise-3.13} \label{sub:exercise-3.13}
Assume that $f$ and $g$ are functions with $f \subseteq g$ and Assume that $f$ and $g$ are functions with $f \subseteq g$ and
@ -3721,7 +3721,7 @@ Show that $f = g$.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.14}}% \subsection{\sorry{Exercise 3.14}}%
\label{sub:exercise-3.14} \label{sub:exercise-3.14}
Assume that $f$ and $g$ are functions. Assume that $f$ and $g$ are functions.
@ -3738,7 +3738,7 @@ Assume that $f$ and $g$ are functions.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.15}}% \subsection{\sorry{Exercise 3.15}}%
\label{sub:exercise-3.15} \label{sub:exercise-3.15}
Let $\mathscr{A}$ be a set of functions such that for any $f$ and $g$ in Let $\mathscr{A}$ be a set of functions such that for any $f$ and $g$ in
@ -3751,7 +3751,7 @@ Show that $\bigcup \mathscr{A}$ is a function.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.16}}% \subsection{\sorry{Exercise 3.16}}%
\label{sub:exercise-3.16} \label{sub:exercise-3.16}
Show that there is no set to which every function belongs. Show that there is no set to which every function belongs.
@ -3762,7 +3762,7 @@ Show that there is no set to which every function belongs.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.17}}% \subsection{\sorry{Exercise 3.17}}%
\label{sub:exercise-3.17} \label{sub:exercise-3.17}
Show that the composition of two single-rooted sets is again single-rooted. Show that the composition of two single-rooted sets is again single-rooted.
@ -3774,7 +3774,7 @@ Conclude that the composition of two one-to-one functions is again one-to-one.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.18}}% \subsection{\sorry{Exercise 3.18}}%
\label{sub:exercise-3.18} \label{sub:exercise-3.18}
Let $R$ be the set Let $R$ be the set
@ -3789,7 +3789,7 @@ Evaluate the following: $R \circ R$, $R \restriction \{1\}$,
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.19}}% \subsection{\sorry{Exercise 3.19}}%
\label{sub:exercise-3.19} \label{sub:exercise-3.19}
Let $$A = \{ Let $$A = \{
@ -3808,7 +3808,7 @@ Evaluate each of the following: $A(\emptyset)$, $\img{A}{\emptyset}$,
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.20}}% \subsection{\sorry{Exercise 3.20}}%
\label{sub:exercise-3.20} \label{sub:exercise-3.20}
Show that $F \restriction A = F \cap (A \times \ran{F})$. Show that $F \restriction A = F \cap (A \times \ran{F})$.
@ -3819,7 +3819,7 @@ Show that $F \restriction A = F \cap (A \times \ran{F})$.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.21}}% \subsection{\sorry{Exercise 3.21}}%
\label{sub:exercise-3.21} \label{sub:exercise-3.21}
Show that $(R \circ S) \circ T = R \circ (S \circ T)$. Show that $(R \circ S) \circ T = R \circ (S \circ T)$.
@ -3830,7 +3830,7 @@ Show that $(R \circ S) \circ T = R \circ (S \circ T)$.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.22}}% \subsection{\sorry{Exercise 3.22}}%
\label{sub:exercise-3.22} \label{sub:exercise-3.22}
Show that the following are correct for any sets. Show that the following are correct for any sets.
@ -3848,7 +3848,7 @@ Show that the following are correct for any sets.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.23}}% \subsection{\sorry{Exercise 3.23}}%
\label{sub:exercise-3.23} \label{sub:exercise-3.23}
Let $I_A$ be the identity function on the set $A$. Let $I_A$ be the identity function on the set $A$.
@ -3862,7 +3862,7 @@ Show that for any sets $B$ and $C$,
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.24}}% \subsection{\sorry{Exercise 3.24}}%
\label{sub:exercise-3.24} \label{sub:exercise-3.24}
Show that for a function $F$, Show that for a function $F$,
@ -3874,7 +3874,7 @@ Show that for a function $F$,
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.25}}% \subsection{\sorry{Exercise 3.25}}%
\label{sub:exercise-3.25} \label{sub:exercise-3.25}
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
@ -3891,7 +3891,7 @@ Show that for a function $F$,
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.26}}% \subsection{\sorry{Exercise 3.26}}%
\label{sub:exercise-3.26} \label{sub:exercise-3.26}
Prove the second halves of parts (a) and (b) of Theorem 3K. Prove the second halves of parts (a) and (b) of Theorem 3K.
@ -3902,7 +3902,7 @@ Prove the second halves of parts (a) and (b) of Theorem 3K.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.27}}% \subsection{\sorry{Exercise 3.27}}%
\label{sub:exercise-3.27} \label{sub:exercise-3.27}
Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$. Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$.
@ -3914,7 +3914,7 @@ Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.28}}% \subsection{\sorry{Exercise 3.28}}%
\label{sub:exercise-3.28} \label{sub:exercise-3.28}
Assume that $f$ is a one-to-one function from $A$ into $B$, and that $G$ is the Assume that $f$ is a one-to-one function from $A$ into $B$, and that $G$ is the
@ -3928,7 +3928,7 @@ Show that $G$ maps $\powerset{A}$ one-to-one into $\powerset{B}$.
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.29}}% \subsection{\sorry{Exercise 3.29}}%
\label{sub:exercise-3.29} \label{sub:exercise-3.29}
Assume that $f \colon A \rightarrow B$ and define a function Assume that $f \colon A \rightarrow B$ and define a function
@ -3943,7 +3943,7 @@ Does the converse hold?
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.30}}% \subsection{\sorry{Exercise 3.30}}%
\label{sub:exercise-3.30} \label{sub:exercise-3.30}
Assume that $F \colon \powerset{A} \rightarrow \powerset{A}$ and that $F$ has Assume that $F \colon \powerset{A} \rightarrow \powerset{A}$ and that $F$ has
@ -3963,7 +3963,7 @@ Define
\end{proof} \end{proof}
\subsection{\unverified{Exercise 3.31}}% \subsection{\sorry{Exercise 3.31}}%
\label{sub:exercise-3.31} \label{sub:exercise-3.31}
Show that from the first form of the axiom of choice we can prove the second Show that from the first form of the axiom of choice we can prove the second

8
DocGen4/Output/Index.lean
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@ -47,11 +47,15 @@ def index : BaseHtmlM Html := do templateExtends (baseHtml "Index") <|
statements, theorems, lemmas, etc. that have been proven in both statements, theorems, lemmas, etc. that have been proven in both
LaTeX and Lean. LaTeX and Lean.
</li> </li>
<li>
<span style="color:olive">Olive statements </span> are reserved for
statements, theorems, lemmas, etc. that have been proven in LaTeX
and will not be proven in Lean.
</li>
<li> <li>
<span style="color:fuchsia">Fuchsia statements </span> are reserved <span style="color:fuchsia">Fuchsia statements </span> are reserved
for definitions, axioms, statements, theorems, lemmas, etc. that for definitions, axioms, statements, theorems, lemmas, etc. that
have been proven or encoded in LaTeX but not yet proven or encoded have been proven or encoded in LaTeX and will be encoded in Lean.
in Lean.
</li> </li>
<li> <li>
<span style="color:maroon">Maroon </span> serves as a catch-all for <span style="color:maroon">Maroon </span> serves as a catch-all for

6
preamble.tex
View File

@ -118,9 +118,11 @@
\texorpdfstring{\color{darkgray}\faParagraph\ #1}{#1}} \texorpdfstring{\color{darkgray}\faParagraph\ #1}{#1}}
\DeclareRobustCommand{\verified}[1]{% \DeclareRobustCommand{\verified}[1]{%
\texorpdfstring{\color{teal}\faCheckCircle\ #1}{#1}} \texorpdfstring{\color{teal}\faCheckCircle\ #1}{#1}}
\DeclareRobustCommand{\partial}[1]{%
\texorpdfstring{\color{Fuchsia}\faPencil*\ #1}{#1}}
\DeclareRobustCommand{\unverified}[1]{% \DeclareRobustCommand{\unverified}[1]{%
\texorpdfstring{\color{olive}\faCheckCircle[regular]\ #1}{#1}}
\DeclareRobustCommand{\pending}[1]{%
\texorpdfstring{\color{Fuchsia}\faPencil*\ #1}{#1}}
\DeclareRobustCommand{\sorry}[1]{%
\texorpdfstring{\color{Maroon}\faExclamationCircle\ #1}{#1}} \texorpdfstring{\color{Maroon}\faExclamationCircle\ #1}{#1}}
% ======================================== % ========================================