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\noindent
\parbox{2.25in}{\bf Math 4710/5710}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Jan. 22${}^\mathbf{nd}$\!, 2025}

%\vspace{0.2in}
%
%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

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{\bf Grads. -- please do both versions --} regarding problems \#1b, \#1c, \#2b, and \#2c.

\vspace{0.2in}

\begin{description}
\item[\#1 Preimages] Let $f:X \to Y$. Preimages preserve inclusions, unions, intersections, and differences of sets. 

For regular versions: Let $B_i \subseteq Y$ where $i=0,1$. 

For grad. versions: Let $B_i \subseteq Y$ where $i \in I$ and $I$ is some arbitrary index set.

{\it Note:} Recall that $x \in f^{-1}(B_i)$ if and only if $f(x) \in B_i$.

\begin{enumerate}[(a)]
\item Show $B_0 \subseteq B_1$ implies $f^{-1}(B_0) \subseteq f^{-1}(B_1)$.

\vspace{0.025in}

\item Show $f^{-1}\left(B_0 \cup B_1\right)  = f^{-1}(B_0) \cup f^{-1}(B_1)$.

\vspace{0.025in}

        {\bf[Grads.] Also show} $f^{-1}\left(\bigcup_{i \in I} B_i\right) = \bigcup_{i \in I} f^{-1}(B_i)$.

\vspace{0.025in}

\item Show $f^{-1}\left(B_0 \cap B_1\right)  = f^{-1}(B_0) \cap f^{-1}(B_1)$.

\vspace{0.025in}

        {\bf[Grads.] Also show} $f^{-1}\left(\bigcap_{i \in I} B_i\right) = \bigcap_{i \in I} f^{-1}(B_i)$.

\vspace{0.025in}
         
\item Show $f^{-1}\left(B_0 - B_1\right)  = f^{-1}(B_0) - f^{-1}(B_1)$.
\end{enumerate}

\vspace{0.15in}

\item[\#2 Images] Let $f:X \to Y$. Images preserve inclusions and unions of sets. 

For regular versions: Let $A_i \subseteq X$ where $i=0,1$. 

For grad. versions: Let $A_i \subseteq X$ where $i \in I$ and $I$ is some arbitrary index set.

{\it Note:} Recall that $x \in f(A_i)$ if and only if there exists some $a \in A_i$ such that $f(a)=x$.

\begin{enumerate}[(a)]
\item Show $A_0 \subseteq A_1$ implies $f(A_0) \subseteq f(A_1)$.

\vspace{0.025in}

\item Show $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$.

\vspace{0.025in}

        {\bf[Grads.] Also show} $f\left(\bigcup_{i \in I} A_i\right) = \bigcup_{i \in I} f(A_i)$.

\vspace{0.025in}

\item $f(A_0 \cap A_1) \subseteq f(A_0) \cap f(A_1)$; show equality holds if $f$ is injective.

\vspace{0.025in}

        {\bf[Grads.] Also show} $f\left(\bigcap_{i \in I} A_i\right) \subseteq \bigcap_{i \in I} f(A_i)$; show equality holds if $f$ is injective.

\vspace{0.025in}

\item $f(A_0 - A_1) \supseteq f(A_0) - f(A_1)$; show equality holds if $f$ is injective.

\vspace{0.025in}

\end{enumerate}

\end{description}

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