

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Fall 1998 Midterm
% Added by George
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\examonly{\newpage}

\problem {\bf (5 points)}  Suppose that we have functions
$f : A \mapsto B$, $g : B \mapsto C$, and $h : C \mapsto A$ such that
for all $a \in A$, $h(g(f(a))) = a$, and for all $b \in B$,
$f(h(g(b))) = b$.  Proofs are not required for any part of this
problem.

\begin{problemparts}
\problempart {\bf (1 points)}  Which of the functions $f$, $g$, and $h$ must be injective?

\examonly{\vspace{1in}}

\solution{
The functions $f$ and $g$ must injective.  }

\problempart {\bf (1 points)}  Which of the functions $f$, $g$, and $h$ must be surjective?

\examonly{\vspace{1in}}

\solution{
The functions $f$ and $h$ must be surjective.  }

\problempart {\bf (2 points)}  Circle the relations that must hold.

\[
\begin{array}{ccccc}
|B| > |C| &\quad& \us{|C| \geq |A|} &\quad& \us{|B| \leq |C|} \\
\\
\us{|A| = |B|}  &\quad& |A| \neq |C|        &\quad& |A| + |B| \geq |C|
\\
\end{array}
\]

\problempart {\bf (1 points)}  If $g$ is surjective, then which of the
functions $f$, $g$, and $h$ must be bijective?

\solution{
If $g$ is surjective, then all of the functions must be bijective.
}

\end{problemparts}