diff --git a/Bookshelf/Apostol/Chapter_1_11.tex b/Bookshelf/Apostol/Chapter_1_11.tex index 6fc243b..61b5c70 100644 --- a/Bookshelf/Apostol/Chapter_1_11.tex +++ b/Bookshelf/Apostol/Chapter_1_11.tex @@ -4,7 +4,7 @@ \newcommand{\link}[1]{\lean{../..} {Bookshelf/Apostol/Chapter\_1\_11} % Location - {Apostol.Chapter\_1\_11.#1} % Namespace + {Apostol.Chapter\_1\_11.#1} % Fragment {Chapter\_1\_11.#1} % Presentation } diff --git a/Bookshelf/Apostol/Chapter_I_03.tex b/Bookshelf/Apostol/Chapter_I_03.tex index f62709c..2cb5c0c 100644 --- a/Bookshelf/Apostol/Chapter_I_03.tex +++ b/Bookshelf/Apostol/Chapter_I_03.tex @@ -4,7 +4,7 @@ \newcommand{\link}[1]{\lean{../..} {Bookshelf/Apostol/Chapter\_I\_03} % Location - {Apostol.Chapter\_I\_03.#1} % Namespace + {Apostol.Chapter\_I\_03.#1} % Fragment {Chapter\_I\_03.#1} % Presentation } diff --git a/Bookshelf/Enderton/Chapter_0.tex b/Bookshelf/Enderton/Chapter_0.tex index 0405113..5d04a36 100644 --- a/Bookshelf/Enderton/Chapter_0.tex +++ b/Bookshelf/Enderton/Chapter_0.tex @@ -3,9 +3,9 @@ \input{../../preamble} \newcommand{\link}[1]{\lean{../..} - {Bookshelf/Enderton/Chapter_0} % Location - {Enderton.Chapter_0.#1} % Namespace - {Chapter_0.#1} % Presentation + {Bookshelf/Enderton/Chapter\_0} % Location + {Enderton.Chapter\_0.#1} % Fragment + {Chapter\_0.#1} % Presentation } \begin{document} diff --git a/Common/Real/Geometry/Area.tex b/Common/Real/Geometry/Area.tex new file mode 100644 index 0000000..4b825b9 --- /dev/null +++ b/Common/Real/Geometry/Area.tex @@ -0,0 +1,97 @@ +\documentclass{article} + +\input{../../preamble} + +\newcommand{\link}[2]{\lean{../..} + {Common/Real/Geometry/Area} % Location + {#1} % Fragment + {#2} % Presentation +} + +\begin{document} + +\header{Axiomatic Framework of Area}{Tom M. Apostol} + +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: + +\section*{\verified{Nonnegative Property}}% +\hyperlabel{sec:nonnegative-property}% + +For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$. + +\begin{axiom} + + \link{Nonnegative-Property}{Nonnegative Property} + +\end{axiom} + +\section*{\verified{Additive Property}}% +\hyperlabel{sec:additive-property}% + +If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in + $\mathscr{M}$, and we have $a(S \cup T) = a(S) + a(T) - a(S \cap T)$. + +\begin{axiom} + + \link{Additive-Property}{Additive Property} + +\end{axiom} + +\section*{\verified{Difference Property}}% +\hyperlabel{sec:difference-property}% + +If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in + $\mathscr{M}$, and we have $a(T - S) = a(T) - a(S)$. + +\begin{axiom} + + \link{Difference-Property}{Difference Property} + +\end{axiom} + +\section*{\verified{Invariance Under Congruence}}% +\hyperlabel{sec:invariance-under-congruence}% + +If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is + also in $\mathscr{M}$ and we have $a(S) = a(T)$. + +\begin{axiom} + + \link{Invariant-Under-Congruence}{Invariance Under Congruence} + +\end{axiom} + +\section*{\verified{Choice of Scale}}% +\label{sec:choice-scale} + +Every rectangle $R$ is in $\mathscr{M}$. +If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$. + +\begin{axiom} + + \link{Choice-of-Scale}{Choice of Scale} + +\end{axiom} + +\section*{\proceeding{Exhaustion Property}}% +\hyperlabel{sec:exhaustion-property}% + +Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so + that + \begin{equation} + \label{sec:exhaustion-property-eq1} + S \subseteq Q \subseteq T. + \end{equation} +If there is one and only one number $c$ which satisfies the inequalities + $$a(S) \leq c \leq a(T)$$ for all step regions $S$ and $T$ satisfying (1.1), + then $Q$ is measurable and $a(Q) = c$. + +\begin{axiom} + + \link{Exhaustion-Property}{Exhaustion Property} + +\end{axiom} + +\end{document} diff --git a/Common/Real/Sequence/Arithmetic.tex b/Common/Real/Sequence/Arithmetic.tex index 190d1f8..dacffd7 100644 --- a/Common/Real/Sequence/Arithmetic.tex +++ b/Common/Real/Sequence/Arithmetic.tex @@ -4,7 +4,7 @@ \newcommand{\link}[1]{\lean{../../..} {Common/Real/Sequence/Arithmetic} % Location - {Real.Arithmetic.#1} % Namespace + {Real.Arithmetic.#1} % Fragment {Real.Arithmetic.#1} % Presentation } diff --git a/Common/Real/Sequence/Geometric.tex b/Common/Real/Sequence/Geometric.tex index 56b1a63..ac23235 100644 --- a/Common/Real/Sequence/Geometric.tex +++ b/Common/Real/Sequence/Geometric.tex @@ -4,7 +4,7 @@ \newcommand{\link}[1]{\lean{../../..} {Common/Real/Sequence/Geometric} % Location - {Real.Geometric.#1} % Namespace + {Real.Geometric.#1} % Fragment {Real.Geometric.#1} % Presentation } diff --git a/preamble.tex b/preamble.tex index e1c36a5..49d6b05 100644 --- a/preamble.tex +++ b/preamble.tex @@ -17,7 +17,7 @@ % the root of the workspace (i.e. where this `preamble.tex` file is located). % #1 - Path to root % #2 - Location -% #3 - Namespace +% #3 - Fragment % #4 - Presentation \newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}} \newcommand{\hyperlabel}[1]{% @@ -28,6 +28,9 @@ % Environments % ======================================== +\newenvironment{axiom}{% + \paragraph{\normalfont\normalsize\textit{Axiom.}}} + {\hfill$\square$} \newcommand{\divider}{% \vspace{10pt} \hrule