178 lines
5.3 KiB
TeX
178 lines
5.3 KiB
TeX
\documentclass{article}
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\usepackage{amsmath}
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\input{../../preamble}
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\newcommand{\larea}[2]{\lean{../..}{Bookshelf/Real/Geometry/Area}{#1}{#2}}
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\newcommand{\lrect}[2]{\lean{../..}{Bookshelf/Real/Geometry/Rectangle}{#1}{#2}}
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\begin{document}
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The properties of area in this set of exercises are to be deduced from the
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axioms for area stated in the foregoing section.
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\section{Exercise 1}%
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\label{sec:exercise-1}
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Prove that each of the following sets is measurable and has zero area:
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\subsection{Exercise 1a}%
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\label{sub:exercise-1a}
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A set consisting of a single point.
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\begin{proof}
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Let $S$ be a set consisting of a single point.
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By definition of a \lrect{Real.Point}{Point}, $S$ is a rectangle in which all
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vertices coincide.
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By \larea{Choice-of-Scale}{Choice of Scale}, $S$ is measurable with area its
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width times its height.
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The width and height of $S$ is trivially zero.
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Therefore $a(S) = (0)(0) = 0$.
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\end{proof}
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\subsection{Exercise 1b}%
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\label{sub:exercise-1b}
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A set consisting of a finite number of points in a plane.
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\begin{proof}
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We show for all $k > 0$, a set consisting of $k$ points in a plane is
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measurable with area $0$.
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\paragraph{Base Case}%
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Consider a set $S$ consisting of a single point in a plane.
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By \eqref{sub:exercise-1a}, $S$ is measurable with area $0$.
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\paragraph{Induction Step}%
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Define our induction hypothesis as, "Let $k > 0$ and assume a set consisting
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of $k$ points in a plane is measurable with area $0$."
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Consider a set $S_{k+1}$ consisting of $k + 1$ points in a plane.
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Pick an arbitrary point of $S_{k+1}$.
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Denote the set containing just this point as $T$.
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Denote the remaining set of points as $S_k$.
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By construction, $S_{k+1} = S_k \cup T$.
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By the induction hypothesis, $S_k$ is measurable with area $0$.
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By \eqref{sub:exercise-1a}, $T$ is measurable with area $0$.
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By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
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measurable, $S_k \cap T$ is measurable, and
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\begin{align}
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a(S_{k+1})
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& = a(S_k \cup T) \nonumber \\
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& = a(S_k) + a(T) - a(S_k \cap T) \nonumber \\
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& = 0 + 0 - a(S_k \cap T). \label{sub:exercise-1b-eq1}
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\end{align}
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\noindent
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There are two cases to consider:
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\subparagraph{Case 1}%
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$S_k \cap T = \emptyset$.
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Then it trivially follows that $a(S_k \cap T) = 0$.
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\subparagraph{Case 2}%
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$S_k \cap T \neq \emptyset$.
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Since $T$ consists of a single point, $S_k \cap T = T$.
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By \eqref{sub:exercise-1a}, $a(S_k \cap T) = a(T) = 0$.
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\vspace{8pt}
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\noindent
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In both cases, \eqref{sub:exercise-1b-eq1} evaluates to $0$, implying
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$a(S_{k+1}) = 0$ as expected.
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\paragraph{Conclusion}%
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By mathematical induction, it follows for all $n > 0$, a set consisting of
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$n$ points in a plane is measurable with area $0$.
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\end{proof}
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\subsection{Exercise 1c}%
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\label{sub:exercise-1c}
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The union of a finite collection of line segments in a plane.
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\begin{proof}
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We show for all $k > 0$, a set consisting of $k$ line segments in a plane is
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measurable with area $0$.
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\paragraph{Base Case}%
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Consider a set $S$ consisting of a single line segment in a plane.
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By definition of a \lrect{Real.LineSemgnet}{Line Segment}, $S$ is a
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rectangle in which one side has dimension $0$.
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By \larea{Choice-of-Scale}{Choice of Scale}, $S$ is measurable with area its
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width $w$ times its height $h$.
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Therefore $a(S) = wh = 0$.
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\paragraph{Induction Step}%
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Define our induction hypothesis as, "Let $k > 0$ and assume a set consisting
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of $k$ line segments in a plane is measurable with area $0$."
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Consider a set $S_{k+1}$ consisting of $k + 1$ line segments in a plane.
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Pick an arbitrary line segment of $S_{k+1}$.
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Denote the set containing just this line segment as $T$.
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Denote the remaining set of line segments as $S_k$.
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By construction, $S_{k+1} = S_k \cup T$.
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By the induction hypothesis, $S_k$ is measurable with area $0$.
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By the base case, $T$ is measurable with area $0$.
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By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
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measurable, $S_k \cap T$ is measurable, and
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\begin{align}
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a(S_{k+1})
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& = a(S_k \cup T) \nonumber \\
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& = a(S_k) + a(T) - a(S_k \cap T) \nonumber \\
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& = 0 + 0 - a(S_k \cap T). \label{sub:exercise-1c-eq1}
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\end{align}
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\noindent
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There are two cases to consider:
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\subparagraph{Case 1}%
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$S_k \cap T = \emptyset$.
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Then it trivially follows that $a(S_k \cap T) = 0$.
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\subparagraph{Case 2}%
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$S_k \cap T \neq \emptyset$.
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Since $T$ consists of a single point, $S_k \cap T = T$.
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By the base case, $a(S_k \cap T) = a(T) = 0$.
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\vspace{8pt}
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\noindent
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In both cases, \eqref{sub:exercise-1c-eq1} evaluates to $0$, implying
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$a(S_{k+1}) = 0$ as expected.
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\paragraph{Conclusion}%
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By mathematical induction, it follows for all $n > 0$, a set consisting of
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$n$ line segments in a plane is measurable with area $0$.
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\end{proof}
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\section{Exercise 2}%
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\label{sec:exercise-2}
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Every right triangular region is measurable because it can be obtained as the
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intersection of two rectangles. Prove that every triangular region is measurable
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and that its area is one half the product of its base and altitude.
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\begin{proof}
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TODO
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\end{proof}
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\end{document}
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