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\begin{document} %(
\baselineskip 24pt

\title{Problem Set 02}
% YOUR NAME GOES HERE under Author
\author{Your Name}
\maketitle

\begin{enumerate} %(

\item What is the negation of the statement ``All women are strong,
all men are handsome, and all children are above average.''?

\item Prove (or disprove) the following statements.  {\em State which
proof technique you used.}
\begin{enumerate} %( 
\item ``The sum of 2 consecutive integers is odd.''
% Answer below this line

\item ``The sum of 5 consecutive integers is odd.''
% Answer below this line

\item ``The sum of 5 consecutive integers is evenly divisible by 5.''
% Answer below this line

\end{enumerate} %)

\item What can we say about sets $A$ and $B$ if:
\begin{enumerate} %( 
\item $A \cup B = A$
% Answer below this line

\item $A - B = \emptyset$
% Answer below this line

\item $\lx|A \cup B\rx| = \lx|A\rx| + \lx|B\rx| - \lx|A \cap B\rx|$
% Answer below this line

\end{enumerate} %)

\item Translate the following Set statements into Logic.  For example,
the proposition $A \cup B \subseteq C$ would be $\forall x \lx(\lx(x \in A
\vee x \in B\rx) \implies x \in C\rx)$.
\begin{enumerate} %( 
\item $\lx(A \cap B\rx) \subseteq E$
% Answer below this line

\item $\lx(A \cap B\rx) = \emptyset$
% Answer below this line

\item $\lx(A \cap B\rx) \subseteq \lx(C - D\rx)$
% Answer below this line

\item $\lx(A \cap B\rx) \cup \lx(C - D\rx) = E$
% Answer below this line

\item $A \cup B \subset C$ (Note the use of $\subset$ instead of $\subseteq$)
% Answer below this line

\end{enumerate} %)

\item BONUS: Prove or disprove that the product of 2 of the following
numbers is non-negative: (The proof must be less than a page to
receive credit.)
\begin{itemize} %(
\item $2^{2342} - 8^{780} + 3^{721}$
\item $\sqrt{2}^{\sqrt{2}^{3138}} - \sqrt{3}^{\sqrt{3}^{1108}}$
\item $999^{888} - 888^{999} + 777^{1020}$
\end{itemize} %)
% Answer below this line

\end{enumerate} %)
\end{document} %)