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\begin{document}

\begin{center}
\Large{CS 191\\}
\large{Homework 1\\}
\end{center}
\begin{flushright}
Your name\\
Your Truman Username\\
September 1, 2010\\
\end{flushright}

\textbf{Problem 2a.}  Verify with a truth table that ``$\text{not }
(A \text{ and }  B)$'' and ``$(\text{not } A) \text{ or }
(\text{not } B)$'' are equivalent.

\textit{Answer:} The truth table appears as follows.

\begin{center}
\begin{tabular}{cc|c|cc}
$A$ & $B$ & $A \text{ and }  B$ & $\text{not }(A \text{ and }  B)$ 
& $(\text{not }A)\text{ or } (\text{not } B)$ \\ \hline
true & true & true & false & false \\
true & false & false & true & true \\
false & true & false & true & true \\
false & false & false & true & true \\
\end{tabular}
\end{center}

Since the last two columns are identical, the two expressions are equivalent.

\textbf{Problem 4a.} Prove the statement, if $x$ and $y$ are even,
then $x+y$ is even.

\textit{Answer:}  Let $x$ and $y$ be any two even integers.  Then they
can be written in the form $x=2m$ and $y=2n$ for some integers $m$ and
$n$.  Therefore, the sum $x+y$ can be written as $x+y=2m+2n=2(m+n)$,
which is an even integer.  QED.

Here are some other symbols which may be helpful.  All of these are in
math (or displaymath) mode.  Also see the symbol
glossary starting on Hein, page 923.

\begin{center}
\begin{tabular}{|c|l|l|}\hline
Symbol & \LaTeX & Meaning \\ \hline\hline
$\leq$ & \verb.\leq. & less than or equal to \\ \hline
$x^y$ & \verb.^. (caret) & superscript \\ \hline
$x_y$ & \verb._. (underscore) & subscript \\ \hline
$\in$ & \verb.\in. & is an element of \\ \hline
$\not\in$ & \verb.\not\in. & is not an element of \\ \hline
$\varnothing$ & \verb.\varnothing. & the empty set \\ \hline
$\neq$ & \verb.\neq. & is not equal to \\ \hline
$\mathbb{N}$ & \verb.\mathbb{N}. & natural numbers \\ \hline
$\subset$ & \verb.\subset. & is a subset of \\ \hline
$\cup$ & \verb.\cup. & union \\ \hline
$\cap$ & \verb.\cap. & intersection \\ \hline
$\{\}$ & \verb.\{\}. & set curly braces \\ \hline
$\langle\rangle$ & \verb.\langle\rangle. & list angles \\ \hline
$\Pi$ & \verb.\Pi. & uppercase Greek letter \\ \hline
$\pi$ & \verb.\pi. & lowercase Greek letter \\ \hline
\end{tabular}
\end{center}

There are zillions more symbols.  It's easy to find them.  For the
square root sign, for example, just google for ``latex square root symbol''.

To make a big union, such as the formulations on Hein page 22, use
displaymath mode with \verb.\[. and \verb.\]..

\[
\bigcup_{i=1}^\infty A_i = A_1 \cup \cdots \cup A_n \cup \cdots
\]

Displaymath mode uses a larger font and sets the material centered and
with space above and below.

\end{document}
