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\begin{document}

\title{Mathematical Writing: Proof Style Guide}


\author{Brian Sherson}

\maketitle

\section{Introduction}

In this class, you will be \emph{writing} compositions that communicate
mathematical ideas. Most of these compositions are mathematical proofs.
Such compositions are so much more than just writing down a series
of mathematical expressions. Indeed, a mathematical proof is intended
to \emph{convince} the reader of the correctness of a claim, and as
such, includes explanations. Like any other composition, the rules
of writing apply, and therefore, when writing a proof, one must abide
by proper grammar, spelling, punctuation, and sentence structure.

However, it is unlikely you have had to write a composition for an
English class that involved communicating mathematical ideas beyond
citing statistics. Nevertheless, mathematical proofs should be written
not only with mathematical correctness in mind, but also as though
you were submitting one in a writing class.


\section{Organize your thoughts/Use paragraphs}

\begin{wrapfigure}{o}{1.5in}%
\includegraphics[height=1.75in]{wall_of_text_1664}\end{wrapfigure}%
Short of not doing the assignment at all, the quickest way to a low
score is to make your writing very difficult to read. In particular,
writing a massive wall of text is the quickest way to keep someone
from reading your writing. In the case of your TA and professor, writing
an unorganized composition will result in a low grade. Keep in mind
your TA and professors are human, and as such, fall victim to impatience.

Instead, a composition, unless very short, should be split up into
multiple paragraphs. A paragraph is a \textbf{section of writing that
focuses on a single idea}. When paragraphs are used properly, it is
easier for a reader to find a place to take a break, take a sip of
tea, come back to your composition, and easily identify where to resume
reading.


\section{Know your audience}

When writing a composition is knowing your audience. If you are writing
a composition with a math professor as your audience, you can leave
out many details such as mundane algebra and computations. However,
if you were writing a composition for a peer, or even someone with
slightly less mathematical background than you, you may need to include
such details. You want to include just enough details so that your
audience can follow along, but not so much that it bores them to tears.

However, for the purpose of this class, while you will ultimately
submit your compositions to your TA or professor, your audience will
be your peers. The goal is that we want you to be able to effectively
communicate mathematical ideas to your peers, not just mathematicians.
Incidentally, it will be highly recommended that you submit your writing
to your peers for feedback, and make revisions as necessary, before
submitting to your TA or professor.


\section{Know what you are writing}

\begin{wrapfigure}{o}{2in}%
\includegraphics[width=2in]{Do-not-think-it-means.jpeg}

\end{wrapfigure}%
Mathematical writing requires precision. As a general rule, you should
know the precise definition of every term you use in a proof. A rather
deflating experience is when a student is asked what ``$d$ divides
$n$'' means, only to receive a response of ``$n$ is divisible
by $d$.'' Such a response is only a restatement of ``$d$ divides
$n$'' into passive voice, and does not actually explain what it
means. Thus, a very important rule to live by when writing mathematics
is that if you do not know the precise meaning of a term, \textbf{do
not use it}.

Another common problem is the abuse of the word ``equation'' and
equal sign ($=$). This comes in the form of students calling everything
that resembles mathematical notation an equation. However, recall
that an equation is a mathematical sentence in which ``$=$'' is
the verb. A related abuse, of course, is using the equal sign as a
generic verb.


\section{Properly introduce all variables}

In almost all cases, any variable that is used should be given a formal
introduction. This is generally done in one of two ways:
\begin{enumerate}
\item When choosing an arbitrary representative of a set: 

\begin{enumerate}
\item Let $\varepsilon>0$...
\item Let $n$ be an even integer...
\item Let $j$ and $k$ be odd integers...
\end{enumerate}
\item When chosen to fulfill a known quality:

\begin{enumerate}
\item Since $k$ is odd, \textbf{there exists an integer $m$ such that}
$k=2m+1$...
\item And so $2=x^{2}$, \textbf{for some positive real number $x$}.
\item Since the function $f$ is blurglecruncheon, \textbf{there exists
a gabbleblotchit function $g$ such that}...
\end{enumerate}
\end{enumerate}
There are notable instances when variables do not need to be explicitly
introduced, but their meaning is understood from their context. Such
notable instances are:
\begin{enumerate}
\item Variables of summation and integration:
\[
\sum_{k=1}^{10}k^{2},\quad\int_{0}^{1}x^{2}\, dx.
\]

\item References to a real vector space of arbitrary dimension: ``Let $G\subseteq\mathbb{R}^{n}$...,''
in which while $G$ is introduced explicitly as an arbitrary subset
of $\mathbb{R}^{n}$, $n$ is understood as being an arbitrary positive
integer.
\end{enumerate}

\section{Sentence Structure}

You may already know from English class that a sentence must contain
a subject and a verb, and in some instances, an object. However, mathematics
is also a language of its own, and has its parts of speech, and sentences.
Consider that ``$x>2$'' is in its own right a sentence, where ``$x$''
is the subject, ``$>$'' is the verb, and ``2'' is the object.

A very important part of doing mathematics is knowing when a symbolic
expression represents a sentence such as ``$x>2$,'' ``$a^{n}+b^{n}\ne c^{n}$,''
and $2\mid6$, and when it represents an object (a noun, if you will),
such as ``$x^{2}+2x+1$.''

Other rules to consider:
\begin{enumerate}
\item Mathematical sentences are never treated as standalone sentences in
a composition, but rather as clauses, functioning as parts of sentences.
For example:

\begin{quotation}
Since $x+2\ge6$, it follows that $x^{2}\ge16$.
\end{quotation}
\item While mathematical sentences can be rewritten entirely in words (e.g.
``$x$ is greater than or equal to three plus $y$,'' in place of
``$x\ge3+y$''), this only makes your composition more difficult
to read, and so you should use the mathematical notation to its fullest.
\item Sentences should never begin with a numeral, nor should they begin
with mathematical notation. Beginning a sentence with a number is
permissible if it is spelled out (``Two,'' ``Ninety-four,'' etc...)
and is for the purpose of counting (used in much the same way you
would in any form of writing.)
\item Simple mathematical notation should be displayed inside of the paragraph
as though it were just another word. Such presentation is called an
in-line formula. For example:

\begin{quotation}
If $x>3$, then $x^{2}>9$.
\end{quotation}
\item Some mathematial notation is more complex, and as such, may be difficult
to display as an in-line formula. To remedy this, such notation should
be displayed on its own line, in the center of the page as follows:

\begin{quotation}
If $ax^{2}+bx+c=0$, and $a\ne0$, then:
\[
x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
\]

\end{quotation}

Notice that the quadratic formula is set in what is called a display
formula. Furthermore, it is essential to remember that this formula
is still considered to be part of a sentence. In this case, the quadratic
formula is the end of the sentence, and so a period must be placed
at the end of the formula.

\item Some formulas can also be set as a display formula if they are generally
regarded as an important formula. Furthermore, it may also be convenient
to number such formulas so as to be able to make references to them
elsewhere in your composition. Such formula numbers can be placed
in either the right or left margins, so long as you maintain a logical
and easy to follow numbering system, and remain consistent.
\item An inevitable part of writing mathematical proofs is a long string
of computations and algebraic manipulations. Such computations and
manipulations should be set as displayed formulas, and each step is
placed on its own separate line, as in:
\begin{flalign*}
 & \hfill & a & =b & \hfill\\
 &  &  & =c\\
 &  &  & =d.
\end{flalign*}
Notice that all the work is done on the right-hand side, and we only
right down the left-hand side on the top line. DO NOT write the left-hand
side on subsequent lines; it will only serve as visual clutter. Also,
notice the equal signs are lined up vertically. However, such manipulations
are not limited to equalities, and in fact, other relations, such
as ``$\le$'' can be mixed in as follows:
\begin{flalign*}
 & \hfill & a & =b & \hfill\\
 &  &  & \le c\\
 &  &  & =d\\
 &  &  & <m\\
 &  &  & =n.
\end{flalign*}
This, for instance, would show that $a<n$.
\item \textbf{Do not cross out} anything to signify any form of cancellation;
just write the new expression on the next line.
\end{enumerate}

\section{Mathematical Statements}

In mathematics, there are four basic classifications of statements:
axioms, definitions, conjectures and theorems. Axioms are basic statements
that form the basis of mathematical knowledge, and are accepted as
true without proof. Definitions are statements that introduce a new
term, and provide its precise meaning. Conjectures are claims for
which there is no proof (and as such, are not admissible for use in
a proof). Theorems are statements that are proven to be true based
on axioms and previously proven theorem. It should be noted that once
a proof of a conjecture is discovered, such conjecture is then considered
to be a theorem.

It should also be noted that the term ``theorem'' itself has a couple
of synonyms: ``lemma'' and ``corollary.'' Use of all three are
often used to distinguish importance. Mathematicians will often reserve
use of the term ``theorem'' for only the most important and famous
of results, for instance, Pythagorean Theorem, Fundamental Theorem
of Calculus, and Fermat's Last Theorem. The term ``lemma'' is then
used to denote a result that is often used as a stepping stone to
a theorem. A corollary is a theorem that is very quickly proven from
a previous theorem, as it is often a special case of such theorem.
Choice of terms is a subjective choice, however, as indicated by the
following quote from Professor Showalter:
\begin{quotation}
``One person's theorem is another person's lemma.''
\end{quotation}
In mathematical writing, it may be important to include such statements.
When including a mathematical statement in a composition, such statement
should begin with a run-in header style label that is set in a different
font style to distinguish it from the actual statement itself. Consider:
\begin{ax*}
For all real numbers $x$ and $y$, $x+y=y+x$.\end{ax*}
\begin{defn*}
Let $n$ be an integer. We say $n$ is odd if there exists an integer
$k$ such that $n=2k+1$.\end{defn*}
\begin{prop*}
For every integer $n$, if $n$ is odd, then $n^{2}$ is also odd.
\end{prop*}
Sometimes, it may be convenient to number mathematical statements
for easy reference:
\begin{ax}
For all real numbers $x$,$y$ and $z$, $x\left(y+z\right)=xy+xz$.\end{ax}
\begin{defn}
Let $n$ be an integer. We say $n$ is even if there exists an integer
$k$ such that $n=2k$.\end{defn}
\begin{prop}
For every integer $n$, if $n^{2}$ is odd, then $n$ is also odd.
\end{prop}
Sometimes, certain mathematical statements are well-known and important
enough to earn themselves a name. Such a name can be included in the
label, set in parenthesis:
\begin{thm}[Intermediate Value Theorem]
Let $f:\left[a,b\right]\rightarrow\mathbb{R}$ be a continuous function.
Then for every $k$ between $f\left(a\right)$ and $f\left(b\right)$,
there exists a $c\in\left[a,b\right]$ such that $k=f\left(c\right)$.
\end{thm}

\section{Proofs}

As stated before, a proof is a piece of writing that has the purpose
of convincing the reader of the correctness of a theorem. When asked
to prove a theorem, you will often be provided with the full statement
of the theorem, and so it will be clear what the end result should
be, although once in a while, you may be tasked with proposing a statement
of a theroem. When assigned a problem set, a cue for when a proof
is appropriate is when such a problem begins with ``Show that...''
or ``Prove that...''
\begin{enumerate}
\item A proof should always be preceeded with a precise statement of the
theorem that is to be proven. 
\item Use run-in header style to denote the beginning of the proof, and
QED or other decoration (usually a square) to denote the end of the
proof.
\item Use \emph{first person plural} pronouns, as if both the author and
reader are proving the result together.
\item Unless method of proof is direct, begin the proof with brief introductory
paragraph that conveys the method of proof that is used.
\item For multipart proofs, use a run-in header to clearly mark each section.

\begin{enumerate}
\item For proofs of biconditional statements ($P\iff Q$), one part is headed
with ``$\implies$'' for the proof of $P\implies Q$, and another
part is headed with ``$\Longleftarrow$'' for the reverse implication
$Q\implies P$. For example:

\begin{prop*}
Let $S$ be a finite set. A function $f:S\rightarrow S$ is one to
one if and only if it is onto.\end{prop*}
\begin{proof}

\begin{subproof}[$\implies$]
Suppose $f$ is one-to-one. We will show $f$ is onto.

...
\end{subproof}


\begin{subproof}[$\Longleftarrow$]
Suppose $f$ is onto. We will show $f$ is one-to-one.

...\qedhere
\end{subproof}
\end{proof}
\item Induction proofs have a ``Basis'' part and ``Induction'' part.

\begin{prop*}
All cars are of the same color.\end{prop*}
\begin{proof}
We will prove this result using induction on the number of cars in
a set.
\begin{subproof}[Basis]
In a set of \textbf{one} car, surely all cars are of the same color.
\end{subproof}


\begin{subproof}[Induction]
Now let $n\in\mathbb{N}$ and suppose all cars in a set of $n$ are
of the same color. We will now show that all cars in a set of $n+1$
are of the same color.

...\qedhere
\end{subproof}
\end{proof}
\item Proofs by cases have parts labeled ``Case 1,'' ``Case 2,'' etc..
\item Each part in a multipart proof should generally begin with an introduction
clearly stating what is being assumed and, with exception to proof
by cases, what is to be proven.
\end{enumerate}
\end{enumerate}
Stuck on how to begin? Consider the following templates for how to
begin the following types of proofs.


\subsection{Direct proof}

A direct proof is a proof of a conditional statement:
\[
\forall x\in U:\underbrace{P\left(x\right)}_{\text{Hypothesis}}\implies\underbrace{Q\left(x\right)}_{\text{Conclusion}}.
\]
In such a proof, we first choose an arbitrary representative of the
domain (``Let $x\in U$...''), and then we assume the hypothesis.
From there, we prove the conclusion. We may begin such a proof utilizing
the following template:
\begin{proof}
Let $x\in U$ and assume $P\left(x\right)$. We will now show $Q\left(x\right)$.

...
\end{proof}
Keep in mind this is a \emph{template}, and as such, we \textbf{do
not just write} ``$P\left(x\right)$'' and ``$Q\left(x\right)$,''
but rather write out the propositional function, as in the following
example:
\begin{prop*}
For each function $f:\mathbb{R}\rightarrow\mathbb{R}$, if $f$ is
globberfluxible, then $f$ is odd.\end{prop*}
\begin{proof}
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function, and assume
$f$ is globberfluxible. We will now show $f$ is odd.

...
\end{proof}

\subsection{Proof by contrapositive}

Once in a while, you may encounter the task of having to prove a conditional
statement:
\[
\forall x\in U:P\left(x\right)\implies Q\left(x\right),
\]
but find that direct proof does not work. In particular, there may
not be an easy logical argument that deduces $Q\left(x\right)$ when
one assumes $P\left(x\right)$.

The good news, however, is that the above conditional statement is
logically equivalent to its contrapositive:
\[
\forall x\in U:\neg Q\left(x\right)\implies\neg P\left(x\right),
\]
and applying the method of direct proof to the contrapositive will
work. This gives the following template:
\begin{proof}
We will prove the contrapositive. Let $x\in U$ and assume $\neg Q\left(x\right)$.
We will now show $\neg P\left(x\right)$.

...
\end{proof}
Keep in mind this is a \emph{template}, and as such, we do not just
write ``$P\left(x\right)$'' and ``$Q\left(x\right)$,'' but rather
write out the actual clauses, as in the following example:
\begin{prop*}
For each $n\in\mathbb{Z}$, if $n$ is globberfluxible, then $n$
is odd.\end{prop*}
\begin{proof}
We will prove the contrapositive. Let $n\in\mathbb{Z}$, and assume
$n$ is even. We will now show $n$ is not globberfluxible.

...
\end{proof}
Notice in this instance that it will be preferred to say ``even''
instead of ``not odd.'' Had there been a term that is precisely
equivalent to ``not globberfluxible,'' it would be preferred that
we use that term instead.


\subsection{Proof by contradiction}

Some statements can only be proven by contradiction, in which it is
assumed the statement is false, with the goal of deducing an absurdity,
either in the form of showing another statement to be both true \emph{and}
false simultaneously, or proving a result that goes against an axiom
or previously-established theorem. Typically, when one writes a contradiction
proof, one uses language that emphasises the fact that one is speaking
hypothetically (``If $\sqrt{2}$ were rational, then there would
exist...'' instead of ``Since $\sqrt{2}$ is rational, there exist...'').

If we were to prove a statement $P$, a proof by contradiction may
begin as follows:
\begin{proof}
Suppose otherwise, that $\neg P$. We will deduce a contradiction.

...
\end{proof}
A famous example of a proof by contradiction is the proof that there
are infinitely many prime numbers.
\begin{proof}
Suppose otherwise, that there are only finitely many primes, say $p_{1}$,
$p_{2}$, ..., $p_{n}$. We will deduce a contradiction.

...
\end{proof}
For those interested, the rest of the proof involves multiplying all
these prime numbers together (something you cannot do to an infinite
number of primes), and go on to eventually show that 1 would have
a prime divisor, something that we know to be false.

It should be noted that applying proof by contradiction to a conditional
statement if often a proof by contrapositive in disguise.


\subsection{Induction}

Proof by inducton is perhaps the most difficult proof method to master,
and the difficulty is generally seen in the opening paragraph of the
Induction part. Recall that the axiom of induction can be written
down in the following symbolic form:
\[
\left(\underbrace{P\left(n_{0}\right)}_{\text{Basis}}\wedge\underbrace{\forall k\ge n_{0}:P\left(k\right)\implies P\left(k+1\right)}_{\text{Induction}}\right)\implies\forall n\ge n_{0}:P\left(n\right).
\]


Note that the Induction part is itself proof of the conditional statement:
\[
\forall k\ge n_{0}:P\left(k\right)\implies P\left(k+1\right),
\]
and as such, the template for direct proofs applies as follows:
\begin{subproof}[Induction]
Let $k\ge n_{0}$ and assume $P\left(k\right)$. We will now show
$P\left(k+1\right)$.
\end{subproof}
What \textbf{not to do} (common mistakes encountered when grading):
\begin{subproof}[Induction]
Assume $P\left(k\right)$ for all $k\ge n_{0}$. We will now show
$P\left(k+1\right)$.
\end{subproof}
Remember to follow the basic rule that $k$ is introduced \emph{first}
as an arbitrary representative of the domain of discourse, \emph{then}
we state the assumption. Assuming ``$P\left(k\right)$ for all $k\ge n_{0}$''
is not the same as assuming $P\left(k\right)$ at an arbitrary choice
of $k$. In fact, ``$P\left(k\right)$ for all $k\ge n_{0}$'' precisely
the statement we are attempting to prove, so we would be begging the
question.

Again, as before, we \emph{write out} what $P\left(k\right)$ and
$P\left(k+1\right)$ are.
\begin{subproof}[Induction]
Let $k=k+1$. We will now show $P\left(k+1\right)$.
\end{subproof}
Writing ``$k=k+1$'' is itself an absurdity. There is \emph{no real
value} of $k$ whatsoever that makes this happen.


\section{Epilogue}

Writing mathematical proofs is not something that is quickly mastered,
but it can be a rewarding experience. Remember that the goal of a
proof is to convince the reader of the correctness of a statement.
Furthermore, mathematics is meaningless without a way to communicate
its ideas, and so it is prudent to learn how to properly integrate
(pun not intended) mathematics with writing. Thus, it is a good idea
to do what is generally encouraged with any piece of writing, including
peer review. After all, you want to make sure your peers completely
understand your writing, as well as your logical argument. Furthermore,
it is always a good idea to read back (aloud) your own writing to
yourself.

Finally, one last piece of advice: Do your computations on scratch
paper first, especially if you are attempting a computation, and are
not sure if it will lead you to your desired result. Once you arrive
at a valid computation that contributes to your proof, then you can
clean up the computation if necessary, and include it in your final
submission.
\end{document}
