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\begin{document}

\begin{center}
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\noindent{\Large{\textsc{Intro to Abstract Mathematics\hfill Assignment 5.2  \\ Spring 2020 \hfill Due February 28}}}

\large

\vspace{1cm}


\begin{exercise}
Let $a, b \in \R$.  Prove that if $a < b$, then $a < \frac{a+b}{2} < b$.
\end{exercise}

\begin{exercise}
Let $x, y, z\in \Z^+$.  Show that if $xy = z$, then $x^2 \le z$ or $y^2 \le z$.
\end{exercise}

\begin{exercise}
\begin{itemize}

\item[\bf a.]  Let $P, Q, R$ be statements.  Prove that $P \rightarrow (Q \vee R) \cong (P \wedge \neg R) \rightarrow Q$.

\item[\bf b.]  Prove that if $n \in \Z$, then $n$ is even or $n+1$ is even.

\end{itemize}
\end{exercise}

\begin{exercise}
Let $f(n) = n^2 + n + 41$.  Prove or disprove the following statement:

\begin{center}
\begin{minipage}{2in}
If $n \in \N$, then $f(n)$ is prime.
\end{minipage}
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\end{exercise}


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