% Anything preceded by a "%" is a _comment_, which is to say,
% something the LaTeX compiler doesn't see or process. This is
% invisible to the LaTeX compiler, so it's useful for writing down
% anything you don't want in your final document (as, for instance, in
% the case of notes you make to yourself, or
% instructions/explanations that I'm providing for your benefit.

% The first line of a LaTeX file is always
% \documentclass[options]{class}
% In the context of this course, this is always going to have the
% option "12pt", which is a request that the standard text be in 12pt
% font, and the class will be "article". There are several different
% classes for different sorts of projects: for instance, there are
% "book" and "letter" classes, but "article" is pretty freeform, so
% it's what we'll use.
\documentclass[12pt]{article}

% Anything beginning with a backslash (like \documentclass) is a
% _command_. Pretty much everything other than literal text in a LaTeX
% document is controlled by commands. Commands may take _arguments_,
% which would either be optional arguments appearing in brackets (like
% "12pt" above) or required arguments. A required argument can be a
% single character, but is more often a set of braces around several
% characters, like {article} above.

% Here we include six packages, which are ways of bringing additional
% functionality into our document.

\usepackage[margin=0.7in]{geometry}

% The "geometry" package gives us better control over margins -- the
% default LaTeX margins are really awfully large -- and we pass it an
% argument to tell it what margin size we want. You can tweak this
% number if you're not happy with the margins, and you could
% explicitly include, say, the options:
% bottom=1in,top=0.5in,left=0.5in,right=0.5in
% if you wanted specific margins on specific sides.

\usepackage{fancyhdr,lastpage}

% fancyhdr is used to put those spiffy headerlines and footerlines on
% the document; we'll configure it later. lastpage is a package which
% calculates the page number of the last page of the document, which
% we'll need to get our "Page X of Y" text on the bottom.

\usepackage{amsmath,amsfonts,amssymb}

% The AMS is the American Mathematical Society, which built a few
% really useful LaTeX packages. amsmath makes a lot of new
% math-formatting capabilities available, amsfonts provides some
% unusual-but-useful fonts like blackboard bold or caligraphic, and
% amssymb provides a number of mathematical symbols.

\pagestyle{fancy}

% This is a command to make LaTeX _use_ the functionality of the
% fancyhdr package. Next up we perform the commands \lhead{...},
% \chead{...}, \rhead{...}, and \lfoot{...}, \cfoot{...}, and
% \rfoot{...} to define the text on the left, center, and right sides
% of the header and footers.

\lhead{MATH 311-01}
\chead{Problem of the Day}
% You'll probably want to change this to your actual name.
\rhead{YOUR NAME HERE}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{\textbf{due September 4, 2013}}

% The text above is mostly self-evident, in that it's literally
% reproduced in the output, but the argument to \cfoot is a bit
% involved. \thepage is a command to include the current page
% number. Commands ordinarily subsume whitespace after them, so that
% "\thepage of" might be rendered "1of". For that reason we also
% include a backslash at the end, to demand that the next space be
% honored. \pageref{...} is a command to write out the page on which a
% specific _label_ appears. Labels are basically defined positions in
% the text, and in this case the label LastPage is defined by
% (surprise!) the package lastpage, which we included above.

% Another comment: the \rfoot argument had the command \textbf{...},
% which was used to typeset its argument in boldface.

\begin{document}

% Everything up to this point has been _frontmatter_, which is to say,
% setup rather than the actual content of our document. Now we start
% the document, using a very common command. This command
% is \begin{environment}, where the environment can be one of many,
% many different types of object. Every \begin in a document needs a
% matched \end. In this case we're beginning the environment
% "document", a pretty special one which every article has exactly
% one of.

\begin{em}
  % "em" is a type of environment. In particular, it stands for
  % "emphasized", which in practice usually means "italicized"
  % (however, emphasizing text that is already italicized will make it
  % upright, so the idea is that emphasized text is _different_ from
  % whatever surrounds it). When I write up solutions to problems, I
  % like to put the problems and provided introductory text in
  % italics, and the solution in upright
  % text.


  % Below we see some mathematics. Inline mathematics is denoted with
  % dollar signs surrounding the mathematical expression. These
  % expressions can be as short or as long as you might like: so, for
  % instance, writing "$A$" might seem silly, but that way it typesets
  % "A" in ways consistent with it being a variable
  % (e.g. italicization, spacing).

  Let $A$, $B$, and $C$, be sets given by $A=\{1,3,5\}$,
  $B=\{n\in\mathbb{Z}:0<n\leq 3\}$, and
  $C=\big\{1,2,3,\{1,3,5\}\big\}$. For each of the following
  questions, briefly explain your reasoning.

  % Note that since "{" and "}" have special meanings to LaTeX,
  % writing a
  % literal brace is a bit tricky. We write "\{" and "\}" to tell
  % LaTeX that this isn't a LaTeX-interpreted brace, but a literal
  % one.

  % In defining B, we used some new symbols. For instance, \in
  % typesets an "is-an-element-of" symbol, \mathbb{...} typesets
  % letters in blackboard bold, and \leq is a "less-than-or-equal-to"
  % symbol.

  % You can put \big, \Big, \bigg, or \Bigg before a delimiter (e.g. a
  % brace, bracket, or parenthesis) to make it larger than
  % usual. There are also ways to make delimiters automatically resize
  % to their contnets which we'll get to later. Here I used \big on
  % the outer braces of the definition of C as a courtesy to the
  % reader, making the outer braces larger.
\end{em}

\begin{enumerate}
  % "enumerate" is an environment for building ordered lists. The
  % outermost list will be numbered; if more "enumerate" environments
  % are nested, they will use different ordered forms (alphabetical,
  % Roman numeral, etc.)

  % Individual items within an enumerated list (or an itemized list,
  % which is an "itemize" environment) are introduced with the \item
  % command.

\item \emph{Calculate $|A|$, $|B|$, and $|C|$.}
  % We use \emph{...} for short emphasized text, which we need for
  % this question. Next, after a blank line, you put your
  % answer. Obviously the answer I have here isn't very good, so you
  % should probably replace it with something better.

  I determined that $|A|=\pi$, because reasons. $|B|=\sqrt 2$,
  because, hey, why not? And $|C|=-14$, because I like $-14$.
    
\item \emph{Is $B\subseteq A$? Is $A\subseteq C$? Is $B\subseteq C$?} 
  % \subseteq typesets a "is a not-necessarily-pro-per-subset-of"
  % symbol. It _only_ works when you're typesetting math, but why
  % would you use it otherwise?

  I think $B\subseteq A$ because blah blah blah. However,
  $B\not\subseteq C$ because hrumf hrumf hrumf, and $A\subseteq C$
  because of psychic intuition.

  % A neat trick above: \not before a relation throws a negation bar
  % around it. There are a few cases where this works slightly
  % suboptimally ("\not =" doesn't typeset as nicely as "\neq", and
  % "\not\in" looks worse than "\notin"), but as a general rule it's a
  % good way to negate a relation.

\item \emph{Are any of these sets \emph{elements} of each other?}
  % Note the nested emphasis, and its effect.

  % Your answer here!

\item \emph{Without explicitly listing out the elements of any power
    sets, find at least one element of $\mathcal P(A)\cap\mathcal
    P(B)$.}
  % \mathcal is like \mathbb, in that it formats its argument in a
  % particular style (caligraphic, in this case). However, you might
  % notice that I didn't use braces! This is because by default, if a
  % command takes an argument, it grabs the next "thing" LaTeX sees as
  % its argument. If there's an expression in braces, it grabs that,
  % but otherwise, it'll grab a single character. So "\mathcal P" and
  % "\mathcal{P}" are the same thing. Leaving the braces off can be
  % dangerous under certain circumstances, but it's often a timesaver.

  % \cap is new here. \cap and \cup are the intersection and union
  % symbol respectively.

\item \emph{Without explicitly listing out the elements of any power
    sets, find at least one element of $\mathcal P(A)-P(B)$.}

  % Your answer here!

\item \textbf{Bonus:} \emph{What, if any, relationship do you believe
    to hold between the two sets $\mathcal P(A)\cap\mathcal P(B)$ and
    $\mathcal P(A\cap B)$? What about $\mathcal P(A)\cup\mathcal P(B)$
    and $\mathcal P(A\cup B)$?}

  % Your answer here!

\end{enumerate}

\end{document}
