Math 250, Meredith College
\begin
Opdag hvorfor 18 millioner mennesker verden rundt stoler på Overleaf med deres arbejde.
% Proof portfolio template for Math 250, Meredith College
% Phillip Andreae
% October 16, 2019
% Students should also consult Overleaf resources and advice I will post on Brightspace.
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% This part is called the preamble.
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\documentclass[12pt]{article} % specifies font size and document type
\usepackage{amsmath, amssymb, amsthm} % these packages are necessary for math formatting
% A percent sign comments the rest of the line out,
% so these lines do not effect what is displayed in your document.
\theoremstyle{definition}
% Here I'm defining some "theorem environments":
\newtheorem*{defn}{Definition}
\newtheorem*{prop}{Proposition}
\newtheorem*{theorem}{Theorem}
% (end of the preamble)
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% The body of the document starts after \begin{document}
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\begin{document}
\title{Math 250 Proof Portfolio}
\author{Your Name}
\date{December 4, 2019}
\maketitle % This displays the title, author, and date that were set above.
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% Definitions
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\section*{Definitions} % The * tells it not to number this section
% You should include here the definition
% of every term that appears in one of your proofs.
% Below are some sample definitions.
\begin{defn}
An integer $x$ is \textit{even}
if $x = 2a$ for some integer $a$.
% I'm putting every mathematical expression, including
% variable names, between dollar signs.
\end{defn}
\begin{defn}
A \textit{rectangle} is a quadrilateral
all of whose angles are right angles.
A \textit{square} is a rectangle
all of whose sides are congruent.
\end{defn}
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% Proofs
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\section{Proof by cases} % It will number the sections without *'s automatically
\begin{prop} % I'm calling this a proposition since it doesn't seem famous or important enough to call it a "theorem".
If $X$ is a square, then $X$ is a rectangle.
\end{prop}
\begin{proof}
Suppose $X$ is a square.
I'm going to demonstrate some of the cool ways
that \LaTeX can format mathematical expressions!
Let $s$ be the side length of $X$
and $A$ be the area of $X$.
Then we have $A = s^2$.
Now suppose that $s$ is an even integer.
By definition, this means that $s = 2a$ for some $a \in \mathbb{Z}$.
Then the area of $X$ is
\begin{equation*}
A = s^2 = (2a)^2 = 4a^2,
\end{equation*}
which we also could have formatted over multiple lines as
\begin{align*}
A &= s^2 \\
&= (2a)^2 \\
&= 4a^2,
\end{align*}
which shows that $A$ is a multiple of $4$,
i.e., $A \equiv 0 \pmod{4}$.
We can format set-builder notation like this:
\begin{equation*}
\{ 2n : n \in \mathbb{Z} \} \subseteq \{ m \in \mathbb{Z} : 4|m \}.
\end{equation*}
Now suppose that the side length of $X$
is a function of $t$, where $t \in \mathbb{R}$.
Then by the chain rule, the derivative of
the area is $\frac{dA}{dt} = 2s \frac{ds}{dt}$,
which we could also have formatted as
$\displaystyle \frac{dA}{dt} = 2s \frac{ds}{dt}$
if we wanted to make the fractions larger.
The net change in the area from $t=t_1$
to $t=t_2$ is
\begin{equation*}
A(t_1) - A(t_2) = \int_{t_1}^{t_2} \frac{dA}{dt} \, dt.
\end{equation*}
Notice how I put a period at the end of that sentence!
I said this was a proof by cases, so let's see how to format lists.
Either $X$ is small or $X$ is large. Let's consider those two cases separately:
\begin{enumerate}
\item Suppose $X$ is small \dots
\item Suppose $X$ is large \dots
\end{enumerate}
Here's how to create a bulleted list:
\begin{itemize}
\item \textit{Case 1:} Suppose $X$ is small \dots
\item \textit{Case 2:} Suppose $X$ is large \dots
\end{itemize}
By the way, since a square is by definition
a rectangle all of whose sides are congruent,
$X$ is also a rectangle.
This completes the proof.
Marvel at how \LaTeX will create the ``end-of-proof-box"
automatically:
\end{proof}
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\section{Congruence modulo $n$}
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\section{Proof by contrapositive}
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\section{Proof by contradiction}
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\section{Induction}
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\section{A proof that a function is bijective}
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\section{The triangle inequality}
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\section{The Pythagorean theorem}
Use brackets to give a theorem a name in parentheses:
\begin{theorem}[Pythagorean theorem]
For every right triangle with legs of lengths $a$ and $b$
and hypotenuse of length $c$,
we have that $a^2 + b^2 = c^2$.
\end{theorem}
Compare to:
\begin{theorem}
This theorem does not have a name
in parentheses since I did not use the brackets.
\end{theorem}
You could cite a source by saying something like:
The idea of the following proof, which is originally due to Euclid,
has been borrowed from \cite{pyth}. % This cites the source I named "pyth" in my bibliography
\begin{proof}
Here's the proof.
\end{proof}
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\section{A proof not from this course}
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% Bibliography
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\begin{thebibliography}{1}
\bibitem{hammack} R. Hammack, \textit{Book of Proof}.
Third edition. 2018.
% I've given this source the name "hammack",
% so I can use the command \cite{hammack} when I want to cite it.
\bibitem{pyth} Wikipedia contributors,
\textit{Wikipedia},
``Pythagorean theorem".
https://en.wikipedia.org/w/index.php?title=
Pythagorean\_theorem \\
\&oldid=918079404.
Online; accessed October 15, 2019.
\end{thebibliography}
\end{document}