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\begin{document}

\begin{center}
{\large{\textsc{Homework 17}}} \\
\textsc{Math 3345 -- Spring 2023 -- 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 8 Exercise 9]}} Let $a, b \in \RR$. Suppose $a \geq 0$ and $b \geq 0$. Prove that:
\begin{enumerate}
\item If $a < b$, then $a^2 < b^2$. [Use basic properties of inequalities.]
\item If $a^2 \leq b^2$, then $a \leq b$. [Do not use square roots. Use part (a).]
\item If $a^2 < b^2$, then $a < b$. [Again, do not use square roots. Use part (b).]
\item If $a < b$, then $\sqrt{a} < \sqrt{b}$. [Use part (c).]
\end{enumerate}

\item \textbf{{[Falkner Section 10 Exercise 1]}} Which of the sets $A$, $B$, $C$, $D$, and $E$ below are the same?
\begin{align*}
A &= \{ 3 \}, \qquad B = \{ 2, 4 \}, \qquad C = \{ x \mid \text{$x$ is prime, $x$ is odd, and $x < 5$} \}, \\
D &= \{ x -1 \mid \text{$x$ is prime, $x$ is odd, and $x < 5$} \}, \qquad E = \big\{ x^2 + 2 \mid x \in \{-1, 1\} \big\}.
\end{align*}
Also, how many distinct sets are named here?
\end{enumerate}

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\begin{center}
\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{{[Falkner Section 8 Exercise 11]}} Let $a, b \in \RR$. Suppose $a > 0$ and $b > 0$. Prove that $\sqrt{a + b} < \sqrt{a} + \sqrt{b}$. [\textsc{hint:} Use Exercise 1 above.]

\item \textbf{{[Falkner Section 8 Exercise 10]}} Let $a, b \in \RR$. Prove that if $0 < a < b$, then $a < \sqrt{ab} < b$.

\item \textbf{{[Falkner Section 4 Exercise 12]}} Show that for each real number $x$, $\pi + x$ is irrational or $\pi + x$ is rational.
\end{enumerate}




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