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\noindent
\parbox{2.25in}{\bf Math 4710/5710-101}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $3^\mathrm{rd}$, 2014}

\vspace{0.4in}

\noindent {\sc\large Total Orders:}  A set $T$ equipped with a relation ``$<$'' is {\bf totally ordered} if ``$<$'' has the following properties:
\begin{description}
   \item[Comparability] For all $a,b \in T$, either $a<b$, $b<a$, or $a=b$.
   \item[Non-reflexive] For all $a \in T$, $a \not< a$.
   \item[Transitive] For all $a,b,c \in T$, if $a<b$ and $b<c$, then $a<c$.
\end{description}

{\it Notes:} A relation which satisfies the non-reflexive and transitive properties (but not necessarily comparability) is called a {\it strict partial order}. 
Next, $b>a$ is  the same as $a<b$. Also, $a \leq b$ means either $a<b$ or $a=b$. Likewise, $a \geq b$ means either $b<a$ or $a=b$.

\vspace{0.3in}

\noindent {\sc\large More Definitions:} Let $T$ be a totally ordered set with order relation $<$. Let $A \subseteq T$. 
\begin{description}
   \item[Lower Bound] $x \in T$ is a {\bf lower bound} for $A$ if $x \leq a$ for all $a \in A$.
   \item[Upper Bound] $x \in T$ is an {\bf upper bound} for $A$ if $x \geq a$ for all $a \in A$.
   \item[Infimum] $x \in T$ 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 $y \in T$ we have $y \leq x$ ($x$ is the largest possible lower bound).
   \item[Supremum] $x \in T$ 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 $y \in T$ 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.
   \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.
\end{description}

\vspace{0.3in}

\noindent {\sc\large Problems -- Hey, we've all got 'em:} Let $T$ be a totally ordered set with order relation $<$. Let $A \subseteq T$.
\begin{enumerate}
   \item Suppose $A$ has an infimum. Show that it's unique (so we can say {\it the} infimum and denote it $\mathrm{inf}(A)$).
   \item Suppose $A$ has a supremum. Show that it's unique (so we can say {\it the} supremum and denote it $\mathrm{sup}(A)$).
   \item Give examples of sets with (a) both supremum and infimum, (b) no supremum or infimum, (c) a supremum but no infimum, and (d) an infimum but no supremum.
   \item {}[{\bf Grad Problem}] Show that the least upper bound property implies the greatest lower bound property. [The converse is true as well and has essentially the same proof with all of the $<$'s flipped to $>$'s.]
\end{enumerate}

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