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\parbox{1.95in}{\bf Math 4710/5710}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{1.95in}{\bf \hfill Due: Wed., Jan. 29${}^\mathrm{th}$\!, 2025}

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

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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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A set $X$ equipped with a relation ``$\leq$'' is {\bf totally ordered} if the relation has the following properties:
\begin{description}
   \item[Reflexive] For all $a \in X$, $a \leq a$.
   \item[Anti-Symmetric] For all $a,b \in X$, if $a \leq b$ and $b \leq a$, then $a=b$.
   \item[Transitive] For all $a,b,c \in X$, if $a \leq b$ and $b \leq c$, then $a \leq c$.
   \item[Comparability] For all $a,b \in X$, either $a \leq b$ or $b \leq a$.
\end{description}

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\noindent
Let $X$ be a totally ordered set with order relation $\leq$. Let $A \subseteq X$. 
\begin{description}
   \item[Lower Bound] $x \in X$ is a {\bf lower bound} for $A$ if $x \leq a$ for all $a \in A$.
   \item[Upper Bound] $x \in X$ is an {\bf upper bound} for $A$ if $x \geq a$ for all $a \in A$.
   \item[Infimum] $x \in X$ is a {\bf greatest lower bound} or {\bf infimum} for $A$ if $x \leq a$ for all $a \in A$ ($x$ is a lower bound) and for any other lower bound $y \in X$ we have $y \leq x$ ($x$ is the largest possible lower bound).
   \item[Supremum] $x \in X$ is a {\bf least upper bound} or {\bf supremum} for $A$ if $x \geq a$ for all $a \in A$ ($x$ is an upper bound) and for any other upper bound $y \in X$ we have $y \geq x$ ($x$ is the smallest possible lower bound).
   \item[Least Upper Bound Property] An ordered set has the {\it least upper bound property} if every non-empty subset that is bounded above has a supremum (in that set).
   \item[Greatest Lowest Bound Property] An ordered set has the {\it greatest lower bound property} if every non-empty subset that is bounded below has an infimum (in that set).
\end{description}

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\begin{description}
\item[\#1 Function ?Fun?] Let $f:X \to Y$ and $g:Y \to Z$ for some sets $X$, $Y$, and $Z$.

\begin{enumerate}[(a)]
\item Suppose both $f$ and $g$ are injective. Show $g \circ f$ is too.
\item Suppose both $f$ and $g$ are surjective. Show $g \circ f$ is too.
\item Suppose $f$ is bijective and let $B \subseteq Y$. In this case, ``$f^{-1}(B)$'' could refer to the preimage (=inverse image) of $B$ under $f$ {\it or} the image of $B$ under $f^{-1}$. Let's denote the preimage of $B$ (under $f$) by $C$ and the image of $B$ (under $f^{-1}$) by $D$. Show $C=D$ (so there is no real ambiguity when $f$ has an inverse).
\end{enumerate}

\item[\#2 Total Order Basics] Let $X$ be totally ordered by $\leq$. Let $A \subseteq X$.

\begin{enumerate}[(a)]
\item Suppose $A$ has an infimum. Show that it is unique (so we can say {\it the} infimum and denote it by $\mathrm{inf}(A)$).

{\it Note:} It is also true (with a very simliar proof) that if $A$ has a supremum, then it must be unique (we denote it by $\mathrm{sup}(A)$).

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\item Give examples of sets with\dots
   \begin{enumerate}[i.]
   \item both supremum and infimum, 
   \item no supremum or infimum, 
   \item a supremum but no infimum, and 
   \item an infimum but no supremum.
   \end{enumerate}

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   \item {}[{\bf Grad Problem}] Show that the least upper bound property implies the greatest lower bound property. 
   
   {\it Note:} The converse is true. This fact has essentially the same proof with all of the $\leq$'s flipped to $\leq$'s.
\end{enumerate}

\end{description}

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