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\begin{document}
\pointsinrightmargin{}

\noindent

\section*{Learning Outcomes}
Upon completing this homework, students should be able to

\begin{itemize}
\item \textbf{Set-up} direct, contradiction, contrapositive, and both
  kinds of inductive proofs.
\item \textbf{Write} a two-colunm direct and contradiction proofs.
\item \textbf{Prove} theorems given type of proof to use.
\item \textbf{Prove} theorems after \emph{choosing} proof type.
\end{itemize}

\section*{Assignment}

\emph{A proof of a proposition is a convincing argument that the
  proposition is true.}

\begin{questions}

\question{} \textbf{Translate} the following into English:
  $\forall n \in \mathbb{Z}^+\;\forall k \in \{\omega : \omega \ddiv n\}\;
  k \ndiv n + 1$
  \begin{solution}[0.0in]
  \end{solution}

\question{} Set up a \emph{direct} proof that 19 is \emph{prime}.
  \begin{parts}
  \part{} State the \emph{definition} of \emph{prime(k)}.
    \begin{solution}[0.0in]
    \end{solution}
  \part{} Give the first line of the direct proof.
    \begin{solution}[0.0in]
    \end{solution}
  \part{} Give the last line of this direct proof.
    \begin{solution}[0.0in]
    \end{solution}
  \end{parts}

\question{} Set up a proof by \emph{contradiction} that $\sqrt{19}$ is
  irrational:
  \begin{parts}
\part{} State the \emph{definition} of \emph{irrational(k)}.
  \begin{solution}[0.0in]
  \end{solution}
\part{} Give the first line of the proof.
  \begin{solution}[0.0in]
  \end{solution}
\part{} State the assumption made \emph{for sake of contradiction}.
\part{} Give the last line of this proof.
  \begin{solution}[0.0in]
  \end{solution}
\end{parts}


\question{} If you are asked to prove that ``The sum of any two
  \emph{rational} numbers is \emph{rational}''
  \begin{parts}
  \part{} What type of proof will you use? Justify your answer in a
    sentence.
    \begin{solution}[0.0in]
    \end{solution}
  \part{} Translate the statement into logic and state the first line
    of the proof.
    \begin{solution}[0.0in]
    \end{solution}
  \end{parts}

\question{} If you are asked to prove that ``The sum of any
  \emph{rational} number and an \emph{irrational} number is
  \emph{irrational}''
  \begin{parts}
  \part{} What type of proof will you use? Justify your answer in a
    sentence.
    \begin{solution}[0.0in]
    \end{solution}
  \part{} Translate the statement into logic and state the first line
    of the proof.
    \begin{solution}[0.0in]
    \end{solution}
  \end{parts}

\question{} If you are asked to prove that ``For all non-negative integers,
  \emph{h}, $19 | h^{19} - h$''
  \begin{parts}
  \part{} What type of proof will you use? Justify your answer in a
    sentence.
    \begin{solution}[0.0in]
    \end{solution}
  \part{} Translate the statement into logic and state the first line
    of the proof.
    \begin{solution}[0.0in]
    \end{solution}
  \end{parts}

\question{} \textbf{Prove}, using \emph{weak induction}, $\forall b
  \in \mathbb{Z}^{+} 3\ddiv b^3 + 6b^2 + 5b - 12$.

\question{} \textbf{Prove} that $\forall n \in \mathbb{Z}^{\ge 0} odd(n^2)
  \limplies odd(n)$.

\question{} \textbf{Prove} by \emph{contradiction} that $\sqrt{19}$ is
  irrational. You may assume
  $\forall z \; 19\ddiv z^2 \limplies 19\ddiv z$.

\question{} \textbf{Prove} that the following closed-form for the summation
  holds for all $n \in \mathbb{Z}^+$:

  $\sum\limits_{i=1}^{n} i^2 - 1 = \frac{n(n+1)(2n+1) - 6n}{6}$
  \begin{solution}[0.0in]
  \end{solution}

\end{questions}

\noindent
\textbf{Submit} your answers electronically, in a commonly readable
format
(\emph{e.g.} \lstinline{.pdf}, \lstinline{.txt}, \lstinline{.docx}),
through BrightSpace.

\end{document}

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