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\begin{center}
{\large{\textsc{Homework 9}}} \\
\textsc{Math 3345 -- Spring 2024 -- Kutler}
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\begin{center}
\textbf{Exercises}
\end{center}

Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}
\item \textbf{{[Falkner Section 4 Exercise 5]}} Let $x$ and $y$ be integers. Prove the following statements.
\begin{enumerate}
\item If $xy$ is even, then $x$ is even or $y$ is even.
\item If $xy$ is odd, then $x$ is odd and $y$ is odd.
\end{enumerate}

\item \textbf{{[Falkner Section 4 Exercise 6]}} Let $a$ be an integer. Use the results of the previous exercise to prove the following statements.
\begin{enumerate}
\item If $a^2$ is even, then $a$ is even.
\item If $a^2$ is odd, then $a$ is odd.
\end{enumerate}

\item \textbf{{[Integers Exercise 1]}} Prove Lemma 4: For any $a \in \ZZ$, $-(-a) = a$.

You may \textbf{only} assume the axioms for the integers and Lemmas 1--3 on the Integers handout. You may find the Additive Inverses axiom and Lemma 2 (Uniqueness of Additive Inverses) particularly useful.

\item \textbf{{[Integers Exercise 2]}} Prove Lemma 7: For any $a, b \in \ZZ$, if $a \cdot b = 0$, then $a = 0$ or $b = 0$.

You may use only the axioms for the integers and Lemmas 1 through 6. [\textsc{hint:} Prove the contrapositive statement. If an integer $x$ is not zero, then by the Trichotomy axiom, either $x \in \NN$ or $-x \in \NN$. Now, consider cases and use the Positive Closure axiom.]

\end{enumerate}

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\textbf{Practice Problems}
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It is strongly recommended that you complete the following problems. There is no need to write up polished, final versions of your solutions (although you may find this a useful exercise). Please do not submit any work for these problems.

\begin{enumerate}
\item \textbf{{[Integers Exercise 3]}} Prove Lemma 9: For any $a, b \in \ZZ$, exactly one of the following is true: (i) $a < b$, or (ii) $a = b$, or (iii) $b < a$.

[\textsc{hint:} You will need to use the Trichotomy axiom, the Distributive Law, and Lemmas 4 and 5.]
\end{enumerate}




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