bookshelf/Exercises/Apostol/Exercises_1_7.tex

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\documentclass{article}
\usepackage{amsmath}
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\usepackage{graphicx}
\usepackage{mathrsfs}
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\input{../../preamble}
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\graphicspath{{./images/}}
\newcommand{\larea}[2]{\lean{../..}{Bookshelf/Real/Geometry/Area}{#1}{#2}}
\newcommand{\lrect}[2]{\lean{../..}{Bookshelf/Real/Geometry/Rectangle}{#1}{#2}}
\begin{document}
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.
\section{Exercise 1}%
\label{sec:exercise-1}
Prove that each of the following sets is measurable and has zero area:
\subsection{Exercise 1a}%
\label{sub:exercise-1a}
A set consisting of a single point.
\begin{proof}
Let $S$ be a set consisting of a single point.
By definition of a \lrect{Real.Point}{Point}, $S$ is a rectangle in which all
vertices coincide.
By \larea{Choice-of-Scale}{Choice of Scale}, $S$ is measurable with area its
width times its height.
The width and height of $S$ is trivially zero.
Therefore $a(S) = (0)(0) = 0$.
\end{proof}
\subsection{Exercise 1b}%
\label{sub:exercise-1b}
A set consisting of a finite number of points in a plane.
\begin{proof}
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Define predicate $P(n)$ as "A set consisting of $n$ points in a plane is
measurable with area $0$".
We use induction to prove $P(n)$ holds for all $n > 0$.
\paragraph{Base Case}%
Consider a set $S$ consisting of a single point in a plane.
By \eqref{sub:exercise-1a}, $S$ is measurable with area $0$.
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Thus $P(1)$ holds.
\paragraph{Induction Step}%
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Assume induction hypothesis $P(k)$ holds for some $k > 0$.
Let $S_{k+1}$ be a set consisting of $k + 1$ points in a plane.
Pick an arbitrary point of $S_{k+1}$.
Denote the set containing just this point as $T$.
Denote the remaining set of points as $S_k$.
By construction, $S_{k+1} = S_k \cup T$.
By the induction hypothesis, $S_k$ is measurable with area $0$.
By \eqref{sub:exercise-1a}, $T$ is measurable with area $0$.
By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
measurable, $S_k \cap T$ is measurable, and
\begin{align}
a(S_{k+1})
& = a(S_k \cup T) \nonumber \\
& = a(S_k) + a(T) - a(S_k \cap T) \nonumber \\
& = 0 + 0 - a(S_k \cap T). \label{sub:exercise-1b-eq1}
\end{align}
There are two cases to consider:
\subparagraph{Case 1}%
$S_k \cap T = \emptyset$.
Then it trivially follows that $a(S_k \cap T) = 0$.
\subparagraph{Case 2}%
$S_k \cap T \neq \emptyset$.
Since $T$ consists of a single point, $S_k \cap T = T$.
By \eqref{sub:exercise-1a}, $a(S_k \cap T) = a(T) = 0$.
\vspace{8pt}
\noindent
In both cases, \eqref{sub:exercise-1b-eq1} evaluates to $0$, implying
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$P(k + 1)$ as expected.
\paragraph{Conclusion}%
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By mathematical induction, it follows for all $n > 0$, $P(n)$ is true.
\end{proof}
\subsection{Exercise 1c}%
\label{sub:exercise-1c}
The union of a finite collection of line segments in a plane.
\begin{proof}
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Define predicate $P(n)$ as "A set consisting of $n$ line segments in a plane
is measurable with area $0$".
We use induction to prove $P(n)$ holds for all $n > 0$.
\paragraph{Base Case}%
Consider a set $S$ consisting of a single line segment in a plane.
By definition of a \lrect{Real.LineSemgnet}{Line Segment}, $S$ is a
rectangle in which one side has dimension $0$.
By \larea{Choice-of-Scale}{Choice of Scale}, $S$ is measurable with area its
width $w$ times its height $h$.
Therefore $a(S) = wh = 0$.
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Thus $P(1)$ holds.
\paragraph{Induction Step}%
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Assume induction hypothesis $P(k)$ holds for some $k > 0$.
Let $S_{k+1}$ be a set consisting of $k + 1$ line segments in a plane.
Pick an arbitrary line segment of $S_{k+1}$.
Denote the set containing just this line segment as $T$.
Denote the remaining set of line segments as $S_k$.
By construction, $S_{k+1} = S_k \cup T$.
By the induction hypothesis, $S_k$ is measurable with area $0$.
By the base case, $T$ is measurable with area $0$.
By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
measurable, $S_k \cap T$ is measurable, and
\begin{align}
a(S_{k+1})
& = a(S_k \cup T) \nonumber \\
& = a(S_k) + a(T) - a(S_k \cap T) \nonumber \\
& = 0 + 0 - a(S_k \cap T). \label{sub:exercise-1c-eq1}
\end{align}
There are two cases to consider:
\subparagraph{Case 1}%
$S_k \cap T = \emptyset$.
Then it trivially follows that $a(S_k \cap T) = 0$.
\subparagraph{Case 2}%
$S_k \cap T \neq \emptyset$.
Since $T$ consists of a single point, $S_k \cap T = T$.
By the base case, $a(S_k \cap T) = a(T) = 0$.
\vspace{8pt}
\noindent
In both cases, \eqref{sub:exercise-1c-eq1} evaluates to $0$, implying
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$P(k + 1)$ as expected.
\paragraph{Conclusion}%
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By mathematical induction, it follows for all $n > 0$, $P(n)$ is true.
\end{proof}
\section{Exercise 2}%
\label{sec:exercise-2}
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 and that its area is one half
the product of its base and altitude.
\begin{proof}
Let $T'$ be a triangular region with base of length $a$, height of length $b$,
and hypotenuse of length $c$.
Consider the translation and rotation of $T'$, say $T$, such that its
hypotenuse is entirely within quadrant I and the vertex opposite the
hypotenuse is situated at point $(0, 0)$.
Let $R$ be a rectangle of width $a$, height $b$, and bottom-left corner at
$(0, 0)$.
By construction, $R$ covers all of $T$.
Let $S$ be a rectangle of width $c$ and height $a\sin{\theta}$, where $\theta$
is the acute angle measured from the bottom-right corner of $T$ relative
to the $x$-axis.
As an example, consider the image below of triangle $T$ with width $4$ and
height $3$:
\begin{figure}[h]
\includegraphics{right-triangle}
\centering
\end{figure}
By \larea{Choice-of-Scale}{Choice of Scale}, both $R$ and $S$ are measurable.
By this same axiom, $a(R) = ab$ and $a(S) = ca\sin{\theta}$.
By the \larea{Additive-Property}{Additive Property}, $R \cup S$ and $R \cap S$
are both measurable.
$a(R \cap S) = a(T)$ and $a(R \cup S)$ can be determined by noting that
$R$'s construction implies identity $a(R) = 2a(T)$.
Therefore
\begin{align*}
a(T)
& = a(R \cap S) \\
& = a(R) + a(S) - a(R \cup S) \\
& = ab + ca\sin{\theta} - a(R \cup S) \\
& = ab + ca\sin{\theta} - (ca\sin{\theta} + \frac{1}{2}a(R)) \\
& = ab + ca\sin{\theta} - ca\sin{\theta} - a(T).
\end{align*}
Solving for $a(T)$ gives the desired identity: $$a(T) = \frac{1}{2}ab.$$
By \larea{Invariance-Under-Congruence}{Invariance Under Congruence},
$a(T') = a(T)$, concluding our proof.
\end{proof}
\section{Exercise 3}%
\label{sec:exercise-3}
Prove that every trapezoid and every parallelogram is measurable and derive the
usual formulas for their areas.
\begin{proof}
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We begin by proving the formula for a trapezoid.
Let $S$ be a trapezoid with height $h$ and bases $b_1$ and $b_2$, $b_1 < b_2$.
There are three cases to consider:
\begin{figure}[h]
\includegraphics[width=\textwidth]{trapezoid-cases}
\centering
\end{figure}
\paragraph{Case 1}%
Suppose $S$ is a right trapezoid.
Then $S$ is the union of non-overlapping rectangle $R$ of width $b_1$ and
height $h$ with right triangle $T$ of base $b_2 - b_1$ and height $h$.
By \larea{Choice-of-Scale}{Choice of Scale}, $R$ is measurable.
By \eqref{sec:exercise-2}, $T$ is measurable.
By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$
are both measurable and
\begin{align*}
a(S)
& = a(R \cup T) \\
& = a(R) + a(T) - a(R \cap T) \\
& = a(R) + a(T) & \text{by construction} \\
& = b_1h + a(T) & \text{Choice of Scale} \\
& = b_1h + \frac{1}{2}(b_2 - b_1)h & \eqref{sec:exercise-2} \\
& = \frac{b_1 + b_2}{2}h.
\end{align*}
\paragraph{Case 2}%
Suppose $S$ is an acute trapezoid.
Then $S$ is the union of non-overlapping triangle $T$ and right trapezoid $R$.
Let $c$ denote the length of base $T$.
Then $R$ has longer base edge of length $b_2 - c$.
By \eqref{sec:exercise-2}, $T$ is measurable.
By Case 1, $R$ is measurable.
By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$
are both measurable and
\begin{align*}
a(S)
& = a(T) + a(R) - a(R \cap T) \\
& = a(T) + a(R) & \text{by construction} \\
& = \frac{1}{2}ch + a(R) & \eqref{sec:exercise-2} \\
& = \frac{1}{2}ch + \frac{b_1 + b_2 - c}{2}h & \text{Case 1} \\
& = \frac{b_1 + b_2}{2}h.
\end{align*}
\paragraph{Case 3}%
Suppose $S$ is an obtuse trapezoid.
Then $S$ is the union of non-overlapping triangle $T$ and right trapezoid $R$.
Let $c$ denote the length of base $T$.
Reflect $T$ vertically to form another right triangle, say $T'$.
Then $T' \cup R$ is an acute trapezoid.
By \larea{Invariance-Under-Congruence}{Invariance Under Congruence},
\begin{equation}
\label{par:exercise-3-case-3-eq1}
\tag{3.1}
a(T' \cup R) = a(T \cup R).
\end{equation}
By construction, $T' \cup R$ has height $h$ and bases $b_1 - c$ and $b_2 + c$
meaning
\begin{align*}
a(T \cup R)
& = a(T' \cup R) & \eqref{par:exercise-3-case-3-eq1} \\
& = \frac{b_1 - c + b_2 + c}{2}h & \text{Case 2} \\
& = \frac{b_1 + b_2}{2}h.
\end{align*}
\paragraph{Conclusion}%
These cases are exhaustive and in agreement with one another.
Thus $S$ is measurable and $$a(S) = \frac{b_1 + b_2}{2}h.$$
\vspace{4pt}
\hrule
\vspace{10pt}
Let $P$ be a parallelogram with base $b$ and height $h$.
Then $P$ is the union of non-overlapping triangle $T$ and right trapezoid $R$.
Let $c$ denote the length of base $T$.
Reflect $T$ vertically to form another right triangle, say $T'$.
Then $T' \cup R$ is an acute trapezoid.
By \larea{Invariance-Under-Congruence}{Invariance Under Congruence},
\begin{equation}
\label{par:exercise-3-eq2}
\tag{3.2}
a(T' \cup R) = a(T \cup R).
\end{equation}
By construction, $T' \cup R$ has height $h$ and bases $b - c$ and $b + c$
meaning
\begin{align*}
a(T \cup R)
& = a(T' \cup R) & \eqref{par:exercise-3-eq2} \\
& = \frac{b - c + b + c}{2}h & \text{Area of Trapezoid} \\
& = bh.
\end{align*}
\end{proof}
\section{Exercise 4}%
\label{sec:exercise-4}
A point $(x, y)$ is called a \textit{lattice point} if both coordinates $x$ and
$y$ are integers.
Let $P$ be a polygon whose vertices are lattice points.
The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of
lattice points inside the polygon and $B$ denotes the number on the boundary.
\subsection{Exercise 4a}%
\label{sub:exercise-4a}
Prove that the formula is valid for rectangles with sides parallel to the
coordinate axes.
\begin{proof}
Let $P$ be a rectangle with width $w$, height $h$, and lattice points for
vertices.
We assume $P$ has three non-collinear points, ruling out any instances of
points or line segments.
By \larea{Choice-of-Scale}{Choice of Scale}, $P$ is measurable with area
$a(P) = wh$.
By construction, $P$ has $I = (w - 1)(h - 1)$ interior lattice points and
$B = 2(w + h)$ lattice points on its boundary.
The following shows the lattice point area formula is in agreement with
$a(P)$:
\begin{align*}
I + \frac{1}{2}B - 1
& = (w - 1)(h - 1) + \frac{1}{2}\left[ 2(w + h) \right] - 1 \\
& = (wh - w - h + 1) + \frac{1}{2}\left[ 2(w + h) \right] - 1 \\
& = (wh - w - h + 1) + (w + h) - 1 \\
& = wh.
\end{align*}
\end{proof}
\subsection{Exercise 4b}%
\label{sub:exercise-4b}
Prove that the formula is valid for right triangles and parallelograms.
\begin{proof}
Let $T'$ be a right triangle with width $w$ and height $h$.
Let $T$ be the triangle $T'$ translated, rotated, and reflected such that the
its vertices are $(0, 0)$, $(0, w)$, and $(w, h)$, where $w \leq h$.
Let $R$ be the overlapping rectangle of width $w$ and height $h$, situated
with bottom-left corner at $(0, 0)$.
There are two cases to consider:
\paragraph{Case 1}%
Suppose $h / w$ is an integral value.
Then there exist $w + 1$ lattice points on $T$'s hypotenuse.
The number of interior lattices points of $T$ is
\begin{align*}
I
& = \frac{1}{2}\left[ (w - 1)(h - 1) - (w - 1) \right] \\
& = \frac{1}{2}\left[ wh - 2w - h + 2 \right].
\end{align*}
The number of boundary lattice points of $T$ is
\begin{align*}
B
& = (w + 1) + h + (w - 1) \\
& = 2w + h.
\end{align*}
Thus
\begin{align*}
I + \frac{1}{2}B - 1
& = \frac{wh - 2w - h + 2}{2} + \frac{2w + h}{2} - 1 \\
& = \frac{wh - 2w - h + 2 + 2w + h - 2}{2} \\
& = \frac{wh}{2}.
\end{align*}
\paragraph{Case 2}%
Suppose $h / w$ is not an integral value.
Then there exist exactly 2 lattice points on $T$'s hypotenuse.
The number of interior lattice points of $T$ is
\begin{align*}
I
& = \frac{1}{2}\left[ (w - 1)(h - 1) \right] \\
& = \frac{1}{2}\left[ wh - w - h + 1 \right].
\end{align*}
The number of boundary lattice points of $T$ is
\begin{align*}
B
& = (w + 1) + h \\
& = w + h + 1.
\end{align*}
Thus
\begin{align*}
I + \frac{1}{2}B - 1
& = \frac{wh - w - h + 1}{2} + \frac{w + h + 1}{2} - 1 \\
& = \frac{wh - w - h + 1 + w + h + 1 - 2}{2} \\
& = \frac{wh}{2}.
\end{align*}
\paragraph{Conclusion}%
These cases are exhaustive and in agreement with one another.
Thus $$a(T) = I + \frac{1}{2}B - 1.$$
We do not prove this formula is valid for parallelograms here.
Instead, refer to \eqref{sub:exercise-4c} below.
\end{proof}
\subsection{Exercise 4c}%
\label{sub:exercise-4c}
Use induction on the number of edges to construct a proof for general polygons.
\begin{proof}
Define predicate $P(n)$ as "An $n$-polygon with vertices on lattice points has
area $I + \frac{1}{2}B - 1$."
We use induction to prove $P(n)$ holds for all $n \geq 3$.
\paragraph{Base Case}%
A $3$-polygon is a triangle.
By \eqref{sub:exercise-4b}, the lattice point area formula holds.
Thus $P(3)$ holds.
\paragraph{Induction Step}%
Assume induction hypothesis $P(k)$ holds for some $k \geq 3$.
Let $P$ be a $(k + 1)$-polygon with vertices on lattice points.
Such a polygon is equivalent to the union of a $k$-polygon $S$ with a
triangle $T$.
That is, $P = S \cup T$.
Let $I_P$ be the number of interior lattice points of $P$.
Let $B_P$ be the number of boundary lattice points of $P$.
Similarly, let $I_S$, $I_T$, $B_S$, and $B_T$ be the number of interior
and boundary lattice points of $S$ and $T$.
Let $c$ denote the number of boundary points shared between $S$ and $T$.
By our induction hypothesis, $a(S) = I_S + \frac{1}{2}B_S - 1$.
By our base case, $a(T) = I_T + \frac{1}{2}B_T - 1$.
By construction, it follows:
\begin{align*}
I_P & = I_S + I_T + c - 2 \\
B_P & = B_S + B_T - (c - 2) - c \\
& = B_S + B_T - 2c + 2.
\end{align*}
Applying the lattice point area formula to $P$ yields the following:
\begin{align*}
& I_P + \frac{1}{2}B_P - 1 \\
& = (I_S + I_T + c - 2) + \frac{1}{2}(B_S + B_T - 2c + 2) - 1 \\
& = I_S + I_T + c - 2 + \frac{1}{2}B_S + \frac{1}{2}B_T - c + 1 - 1 \\
& = (I_S + \frac{1}{2}B_S - 1) + (I_T + \frac{1}{2}B_T - 1) \\
& = a(S) + (I_T + \frac{1}{2}B_T - 1) & \text{induction hypothesis} \\
& = a(S) + a(T). & \text{base case}
\end{align*}
By the \larea{Additive-Property}{Additive Property}, $S \cup T$ is
measurable, $S \cap T$ is measurable, and
\begin{align*}
a(P)
& = a(S \cup T) \\
& = a(S) + a(T) - a(S \cap T) \\
& = a(S) + a(T). & \text{by construction}
\end{align*}
This shows the lattice point area formula is in agreement with our axiomatic
definition of area.
Thus $P(k + 1)$ holds.
\paragraph{Conclusion}%
By mathematical induction, it follows for all $n \geq 3$, $P(n)$ is true.
\end{proof}
\subsection{Exercise 5}%
\label{sub:exercise-5}
Prove that a triangle whose vertices are lattice points cannot be equilateral.
[\textit{Hint:} Assume there is such a triangle and compute its area in two
ways, using Exercises 2 and 4.]
\begin{proof}
Proceed by contradiction.
Let $T$ be an equilateral triangle whose vertices are lattice points.
Assume each side of $T$ has length $a$.
Then $T$ has height $h = (a\sqrt{3}) / 2$.
By \eqref{sec:exercise-2},
\begin{equation}
\label{sub:exercise-5-eq1}
\tag{5.1}
a(T) = \frac{1}{2}ah = \frac{a^2\sqrt{3}}{4}.
\end{equation}
Let $I$ and $B$ denote the number of interior and boundary lattice points of
$T$ respectively.
By \eqref{sec:exercise-4},
\begin{equation}
\label{sub:exercise-5-eq2}
\tag{5.2}
a(T) = I + \frac{1}{2}B - 1.
\end{equation}
But \eqref{sub:exercise-5-eq1} is irrational whereas
\eqref{sub:exercise-5-eq2} is not.
This is a contradiction.
Thus, there is \textit{no} equilateral triangle whose vertices are lattice
points.
\end{proof}
\subsection{Exercise 6}%
\label{sub:exercise-6}
Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all
subsets of $A$.
(There are 32 altogether, counting $A$ itself and the empty set $\emptyset$.)
For each set $S$ in $\mathscr{M}$, let $n(S)$ denote the number of distinct
elements in $S$.
If $S = \{1, 2, 3, 4\}$ and $T = \{3, 4, 5\}$, compute $n(S \cup T)$,
$n(S \cap T)$, $n(S - T)$, and $n(T - S)$.
Prove that the set function $n$ satisfies the first three axioms for area.
\begin{proof}
Let $S = \{1, 2, 3, 4\}$ and $T = \{3, 4, 5\}$.
Then
\begin{align*}
n(S \cup T)
& = n(\{1, 2, 3, 4\} \cup \{3, 4, 5\}) \\
& = n(\{1, 2, 3, 4, 5\}) \\
& = 5. \\
n(S \cap T)
& = n(\{1, 2, 3, 4\} \cap \{3, 4, 5\}) \\
& = n(\{3, 4\}) \\
& = 2. \\
n(S - T)
& = n(\{1, 2, 3, 4\} - \{3, 4, 5\}) \\
& = n(\{1, 2\}) \\
& = 2. \\
n(T - S)
& = n(\{3, 4, 5\} - \{1, 2, 3, 4\}) \\
& = n(\{5\}) \\
& = 1.
\end{align*}
We now prove $n$ satisfies the first three axioms for area.
\paragraph{Nonnegative Property}%
$n$ returns the length of some member of $\mathscr{M}$.
By hypothesis, the smallest possible input to $n$ is $\emptyset$.
Since $n(\emptyset) = 0$, it follows $n(S) \geq 0$ for all $S \subset A$.
\paragraph{Additive Property}%
Let $S$ and $T$ be members of $\mathscr{M}$.
It trivially follows that both $S \cup T$ and $S \cap T$ are in
$\mathscr{M}$.
Consider the value of $n(S \cup T)$.
There are two cases to consider:
\subparagraph{Case 1}%
Suppose $S \cap T = \emptyset$.
That is, there is no common element shared between $S$ and $T$.
Thus
\begin{align*}
n(S \cup T)
& = n(S) + n(T) \\
& = n(S) + n(T) - 0 \\
& = n(S) + n(T) - n(S \cap T).
\end{align*}
\subparagraph{Case 2}%
Suppose $S \cap T \neq \emptyset$.
Then $n(S) + n(T)$ counts each element of $S \cap T$ twice.
Therefore $n(S \cup T) = n(S) + n(T) - n(S \cap T)$.
\subparagraph{Conclusion}%
These cases are exhaustive and in agreement with one another.
Thus $n(S \cup T) = n(S) + n(T) - n(S \cap T)$.
\paragraph{Difference Property}%
Suppose $S, T \in \mathscr{M}$ such that $S \subseteq T$.
That is, every member of $S$ is a member of $T$.
By definition, $T - S$ consists of members in $T$ but not in $S$.
Thus $n(T - S) = n(T) - n(S)$.
\end{proof}
\end{document}