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\begin{center}\textbf{Problem Set 4 \hspace{.7in} Foundations \hspace{.7in} Due 2/19/24} \end{center}

\noindent Four of these problems will be graded (our 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 you  must write up your own final submission without reference to any sources other than the textbook and instructor. \\

\begin{enumerate}

\item	Do the Set Theory Gateway on WeBWorK, available via
\begin{center} https://webwork.coe.edu/webwork2/MTH-215/ \end{center}

\item For any sets $A, B,$ and $C$, $(A-B) \subseteq (A-C) \cup (C-B)$.
\item Show that $A \cap \displaystyle \left(\bigcup_{i \in I} B_i\right)' = \displaystyle \bigcap_{i \in I} \ (A \cap B_i')$.
\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 $a < b$ and $c < d$.  Then $a \cdot c < b \cdot d$.
\item Suppose that $a, b, c, d \in \mathbb R$, with $a < b$ and $c < d$.  Then $\frac{a}{c} < \frac{b}{d}$.
\item Suppose that $a, b \in \mathbb R$, with $a < b$ and $a,b > 0$. Then $a^2 < b^2$.
\item Suppose that $a, b \in \mathbb R$, with $a < b$ and $a, b > 0$.  Then $\forall n \in \mathbb N, \ a^n \leq b^n$.
\item For all $x, y, z \in \mathbb{R}$, $|x+y+z| \leq |x| + |y| + |z|$.
\item For all $x,y \in \mathbb{R}$, $d(x,y) = 0 \Leftrightarrow x=y$.


\end{enumerate}

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