\documentclass[12pt]{article}

% Everything after a percent sign is ignored by the LaTeX compiler --
% these are called "comments". I will use comments to write up notes
% in the templates to make it easier for you to understand what I am
% doing and, where necessary, mimic it. You can use them to write up
% your own non-compiling notes, or if you have some prepared work
% which you don't want to appear in your final writeup but don't weant
% to delete outright, you can just throw percent signs on the
% beginning of every line to make it "invisible".

% Note that most of this template is comments, to explain exactly what
% is going on with each line. You can remove as many of them as you
% want without affecting the output, and you might well want to remove
% them if you think they're just in your way.

% If you actually want to put a percent sign in your document, use the
% two-character combination "\%", e.g. "40 dollars invested at a 4\%
% interest rate will become $40e^{0.04t}$ dollars after $t$ years."


% Packages add functionality to your document, and are loaded with the
% "\usepackage" command. A few of the standard ones are included here,
% to make your document look better and to give you access to a full
% range of symbols. It is possible some of the following packages
% aren't part of your LaTeX installation; you can remove the
% references to them as necessary, although it would be better to
% install them.

% The "geometry" package is used for page layout. The normal LaTeX
% page is very small; using this package allows us to let out the
% margins a bit. If you find the text unattractively close to the
% edges of the page, you might try adjusting the 'margin=' number to
% 1in. Feel free to experiment!

\usepackage[margin=0.7in]{geometry}

% amsmath, amsfonts, and amssymb are packages created by the American
% Mathematical Society to respectively enhance LaTeX's basic equation
% layout, fonts, and available symbols for relations and operators. Of
% particular note at this point: amsfonts provides the commands
% \mathcal and \mathbb for caligraphic and blackboard-bold text,
% respectively.

\usepackage{amsmath,amsfonts,amssymb}

% fancyhdr allows a wider range of headers and footers than basic
% LaTeX.

\usepackage{fancyhdr}

% lastpage may not be part of your installation; it's a somewhat
% nonstandard package that determines the page number of the last page
% of your document, and allows you to refer to it with
% \pageref{LastPage}. If you don't have it, it can be worked around.

\usepackage{lastpage}

% Uncomment the following line and actually put your name into the
% second pair of braces. This will make LaTeX resolve \studentname
% throughout the document as whatever you put in the braces. Note that
% if you don't uncomment this line, your source will not compile!

%\newcommand{\studentname}{Your name here}

% Here we _use_ the functionality provided by the fancyhdr
% package. The first line tells LaTeX that we actually want a page
% with headers and footers, and the subsequent 6 commands indicate
% what those headers and footers are. Feel free to fiddle with these
% to produce the layout _you_ like!
\pagestyle{fancy}

% In the upper left, we have the class name...
\lhead{MATH 311-02}

% In the upper middle, the title of this particular document...
\chead{Problem Set \#1}

% And in the upper right, the name of the student doing this
% work. Note that this next line will fail to compile unless you
% uncomment the \newcommand up above!
\rhead{\studentname}

% I leave the lower left blank. Maybe you can come up with something
% clever to do with it.
\lfoot{}

% In the lower middle, the page number. The basic version of this uses
% the lastpage package. If you're not using lastpage, then comment
% this one out and use the simpler version which is commented out
% below. Note that \thepage is a macro meaning "the current page
% number".

\cfoot{Page \thepage\ of \pageref{LastPage}}

% In the lower right, the current date. \today will be expanded on
% compilation into whatever day your computer believes it to be.
\rfoot{\today}

% And we haven't even written down an actual word of text yet! But at
% least our page is set up. Now we have the actual document.

% "document" is an example of an environment. Most LaTeX structures
% are either commands (which begin with backslashes, and may take
% parameters) or environments. We've seen several examples of commands
% in the setup above -- \thepage and \today were commands which didn't
% take any parameters, while \lfoot and \pagestyle each take a single
% parameter. An environment is a pair of commands:
% a \begin{environment} command and an \end{environment} command. The
% "document" environment is just what it sounds like: an environment
% thrown around all the actual text of the document. Note that way
% down at the bottom of this file, we'll have an \end{document} to
% match our \begin{document} here.


% One thing that didn't fit elsewhere, and is particularly relevant
% when talking about sets: if we use braces for grouping in LaTeX,
% how do we actually typeset a brace? Precede them with backslashes,
% e.g. \{ and \}

\begin{document}

% Here's another kind of environment -- a numbered list, which is an
% environment called "enumerate". Within the enumerate environment,
% we'll have several "\item"s -- those are commands indicating a
% single item in a numbered list.
\begin{enumerate}
  % \textbf is a formatting command, saying "typeset the argument in
  % boldface (or 'bf' for short)". Here I use it to put the number of
  % points the problem is worth in boldface.

  % I also use \emph, which is similar but instead of boldface, it
  % means "emphasize". Text is emphasized by italicizing it -- or by
  % de-italicizing it! \emph when used within italiciazed text will
  % straighten it. I put the actual problem statement in emphasized
  % text, so that the question, italicized, will appead before your
  % answer. You can remove or edit this if you prefer some other
  % format.
\item \textbf{(10 points)} \emph{We observed in class that when $A$
    is a finite set, $|\mathcal P(A)|=2^{|A|}$. Explain in your own
    words why this is true.}

  % Some new symbols and commands in the above text! Particularly of
  % note are the dollar signs. You throw a pair of dollar signs around
  % anything you want to be typeset inline as math -- with variables
  % in italic, with equalities spaced correctly, and so forth. So $A$
  % will italicize the A properly (\textit{A} would also work, but
  % would be semantically unsound, suggesting "A" is a text element,
  % not a math element). Most mathematical symbols are just where they
  % are on the keyboard, and in the second bit of math we see absolute
  % value signs and equals signs used in unsurprising ways. What's a
  % bit peculiar is the caret, used to provide a superscript: the
  % caret takes a parameter, which is enclosed in braces, which is the
  % thing to be exponentiated. Also, here we use \mathcal to get a
  % script P.

  % If you want a literal dollar sign in your text, use \$, e.g. "if
  % we invest \$40 at 5\%..."

  % A digression: why do we write \mathcal P instead of \mathcal{P}?
  % We _could_ use \mathcal{P}, but \mathcal P also works. It's in how
  % LaTeX processes parameters: the next "thing" after a command that
  % takes a parameter will be used as its parameter, and a "thing" can
  % be a character, or an expression in braces, so \mathcal grabs just
  % the single character P. This behavior can sometimes catch you in
  % surprising ways: Try typesetting $x^2$ and it looks fine, but
  % $x^10$ doesn't look at all right; compare $x^{10}$; when LaTeX
  % grabs just the next thing, it processes $x^10$ as $x^{1}0$, which
  % isn't usually what you mean. If you're paranoid, you can always
  % use braces to make sure parameters are what you want them to be,
  % but you can leave them off when the parameter's a single
  % character.

\item \textbf{(5 points)} \emph{Explain why, when $A\subseteq B$ and
    $A$ and $B$ are both finite sets, it follows that $|A|\leq|B|$.}

  % A few more new symbols: \subseteq is a symbol for a
  % subset-with-line-under-it; \leq is a symbol for less-than-or-equals.

  % Note that commands often need space after them: $A\subseteqB$
  % would crash, because "\subseteqB" isn't the name of a
  % command. $|A|\leq|B|$ is OK without the space, because commands
  % only contain alphabetical characters, so the \leq is interpreted
  % as ending before the vertical line.

\item \textbf{(4 points)} \emph{Give examples of sets satisfying each
    of the conditions below:}

  % Note: an enumerate appearing inside another enumerate will use a
  % different enumeration technique. The outer enumerate environment
  % used Arabic numbers; this one will use lowercase letters.
  \begin{enumerate}
  \item \emph{$S\subseteq\mathcal P(\mathbb N)$.}
    % \mathbb N renders a blackboard-bold N. If you prefer a more
    % traditional bold, you can use \mathbf N.

  \item \emph{$T\in\mathcal P(\mathbb N)$.}
    % \in is a the "is an element of" symbol.

  \item \emph{$A\subseteq\mathcal P(\mathbb N)$ and $|A|=5$.}

  \item \emph{$B\in\mathcal P(\mathbb N)$ and $|B|=5$.}
  \end{enumerate}

\item \textbf{(6 points)} \emph{Explain why, for any sets $A$ and $B$,
    it must always be the case that $A\cap B\subseteq A\subseteq A\cup
    B$. Are there any situations where $A\cap B$ is not a proper
    subset of $A$, or where $A$ is not a proper subset of $A\cup B$.}
  % \cup and \cap are the union and intersection symbols.

\item \textbf{(6 points)} \emph{For each real number $r$, define
    $A_r=\{r^2\}$, define $B_r$ as the closed interval $[r-1,r+1]$,
    and define $C_r$ as the open interval $(r,\infty)$. For
    $S=\{1,2,4\}$, evaluate the following expressions:}
  % \infty is the infinity symbol. Notice how here we used \{ and \}
  % to get real, literal typeset braces instead of LaTeX grouping
  % symbols.
  \begin{enumerate}
  \item \emph{$\bigcup_{\alpha\in S}A_\alpha$ and $\bigcap_{\alpha\in
        S}A_\alpha$.}
    % \bigcup and \bigcap typeset the large 'iteration operator' forms
    % of \cup and \cap.

  \item \emph{$\bigcup_{\alpha\in S}B_\alpha$ and $\bigcap_{\alpha\in
        S}B_\alpha$.}

  \item \emph{$\bigcup_{\alpha\in S}C_\alpha$ and $\bigcap_{\alpha\in
        S}C_\alpha$.}
  \end{enumerate}

\item \textbf{(4 points)} \emph{Find an indexed collection of distinct
    sets $\{A_n\}_{n\in\mathbb N}$ (so that no two sets are equal)
    satisfying the following two conditions: $$\bigcap_{n=1}^\infty
    A_n=\{-1,0,1\}\text{ and }\bigcup_{n=1}^\infty A_n=\mathbb Z$$}
  % There are some unfamiliar bits in the above line! For a start, we
  % have the "$$...$$ formulation, which instead of putting math
  % inline, centers it and displays it larger. It also puts subscripts
  % and superscripts on the iteration operators above and below them,
  % instead of off to the side. We also see \text{...}, which is a
  % good way of putting ordinary nonmathematical text in a line of
  % math.

\item \textbf{(5 points)} \emph{Give an example of a partition of
    $\mathbf Q$ into three subsets.}

  % You only need to do the following if you want to -- which is why
  % it's commented out in the basic template, so that it doesn't show
  % up unless you uncomment it and work on it.
  
%\item \textbf{(5 point bonus)} \emph{I briefly discussed
%    self-reference as a problematic issue in class: here we can look
%    at what makes it a problem. Let us consider, hypothetically, the
%    concept of a set $A$ containing all sets. Since $A$ is itself a
%    set, it would be the case that $A\in A$ (it would also be the case
%    that $\emptyset\in A$ and $\mathcal P(A)\in A$, for those are both
%    sets too).
%
%  So far this is not a problem. But now let us consider $S=\{X\in
%  A:X\notin X\}$. Clearly, for example, $A\notin S$, because as we saw
%  above, $A\in A$. On the other hand, for instance, $\emptyset$ and
%  $\mathbb Z$ would be in $S$, since neither the empty set nor the
%  integers have themselves as members.
%  % Note: \notin is a not-a-member-of sign.
%
%  The key question: is it true or false that $S\in S$, and what would
%  investigating this question tell us?}
\end{enumerate}

\end{document}