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\title{CSCI 301 - Assignment 2, Fall 2024}
\author{Your name here}
\date{}

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\maketitle

\noindent
Modify the .tex source file for this document, replacing the placeholders with
your solutions. \\
This is an individual assignment. See the syllabus for the collaboration policy.\\

\hrule

% Beginning of Questions
\begin{enumerate}
\item (2 points) Give the simplest expression you can find that is equivalent to
$\overline{\overline{A} \cup \overline{B}}$. You may find it helpful to use
Venn diagrams; you may, but aren't required to include them here.
\answer
\emph{Replace this text, including the emphasis, with your answer.}

\item (4 points) Find a way to express $P \Rightarrow Q$ without using $\Rightarrow$;
ideally your expression should be as simple as possible. Verify the equivalence
using either truth tables or known equivalencies (see \emph{Book of Proof}
Section 2.6, page 52). If you use known equivalencies, point out what
equivalency you apply for each step.
\answer
\emph{Replace this text, including the emphasis, with your answer.}

\item Verify each of the following equivalencies in the same manner
as in the prior question. 
\begin{enumerate}
    \item (2 points) $A\Rightarrow B \equiv (A \land \neg B)\Rightarrow False.$
    \answer
    \emph{Replace this text, including the emphasis, with your answer.}

    \item (2 points) $(A\land B)\Rightarrow C \equiv (A\Rightarrow C) \lor (B\Rightarrow C).$
    \answer
    \emph{Replace this text, including the emphasis, with your answer.}
\end{enumerate}


\item For each of the following sentences, do the following three things: (i)
Translate it into a symbolic logic form; (ii) Negate the logic form and then
simplify it as much as you can; (iii) Translate the negated form back to a
sensible English sentence. Define the symbols before you use them in step 1. As
a shorthand for your symbolic representation, you can write ``$E(x)$'' as the
statement that $x$ is even, and $O(x)$ to say $x$ is odd.
\begin{enumerate}
    \item (2 points) The integer a and the integer b are not both odd.
    \answer
    \emph{Replace this text, including the emphasis, with your answer.}
    \item (4 points) Given any two integers a and b, if both ab and a+b are even, then
    both a and b are even. 
    \answer
    \emph{Replace this text, including the emphasis, with your answer.}
    \item (4 points) There exists an integer $x$ for which $x + y = 0$ for any integer $y$.
    \answer
    \emph{Replace this text, including the emphasis, with your answer.}
\end{enumerate}

\end{enumerate}

\section*{Possibly Helpful Latex Things}
\textbf{Remove this section in your submission.} Here are some
examples of \LaTeX~syntax that may help you in answering some of the above
questions.

\noindent A truth table:

\begin{tabular}{|c|c|c|} % 3 centered columns, with vertical separators
\hline % horizontal border / row separator
$P$ & $Q$ & $P \Rightarrow Q$ \\ \hline % table row
$T$ & $T$ & $T$               \\ \hline
$T$ & $F$ & $F$               \\ \hline
$F$ & $T$ & $T$               \\ \hline
$F$ & $F$ & $T$               \\ \hline
\end{tabular}

\noindent An enumerated list:
\begin{enumerate}
    \item Thing 1
    \item Thing 2
    \item Thing 3
\end{enumerate}

\noindent An aligned sequence of equations, one per line:
\begin{align*}
y &= mx + b \\ % lines will be aligned on the location of the & character(s)
y - b &= mx \\ % in this case, we line up the = signs
\frac{y - b}{m} &= x
\end{align*}

\noindent If you want to include an image, see the commented-out syntax below.
% filename.png should be in the same directory as your tex file
% \includegraphics[]{filename.png} 

\end{document}

