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\begin{center}\textbf{Problem Set 5 \hspace{.7in} Foundations \hspace{.7in} Due 2/21/22} \end{center}

\noindent Four of these problems will be graded (my choice, not yours!), with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. Submit as a pdf on Moodle.

\begin{enumerate}
\item For each $x \in \mathbb R^+$, let $Q_x = \{y | \frac{1}{x} < y < \frac{2x+1}{x} \}$.  What are:
    \begin{enumerate}
    \item $$\bigcup_{x \in \{1,2,3 \}} Q_x$$
    \item $$\bigcap_{x \in \{1,2,3 \}} Q_x$$
    \item $$\bigcup_{x \in \mathbb Z^+} Q_x$$
    \item $$\bigcap_{x \in \mathbb Z^+} Q_x$$
    \item $$\bigcup_{x \in \mathbb R^+} Q_x$$
    \item $$\bigcap_{x \in \mathbb R^+} Q_x$$
    \end{enumerate}
\item Show that $\displaystyle \left ( \bigcup_{i \in I} A_i \right ) ^\prime = \displaystyle \bigcap_{i \in I} \ A_i ^\prime$.
\item Show that $A \cap \displaystyle \bigcup_{i \in I} B_i = \displaystyle \bigcup_{i \in I} \ (A \cap B_i)$.
\item Suppose that $a, b, c \in \mathbb R$, and $a-c < b-c$.  Then $a < b$.
\item Suppose that $a, b, c, d \in \mathbb R$, with $a < b$ and $c < d$.  Then $a - d < b - c$.
\item Suppose that $a, b, c, d \in \mathbb R$, with $a < b$, $c < d$, and $a,c>0$.  Then $a \cdot c < b \cdot d$.
\item Suppose that $a, b \in \mathbb R$.  If $a < b$ then $a^2 < b^2$.
\item Suppose that $a, b \in \mathbb R$, with $a < b$ and $a, b > 0$.  Then $\forall n \in \mathbb N, a^n \leq b^n$.


\end{enumerate}

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