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% Semester, course, title
\def\semester{Spring 2014 }
\def\course{Math 2710 }
\def\title{Homework 9 (Due Tues, Apr 8)}

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\rhead{First and Last Name} % enter your name here


\begin{document}

% Header
\vspace*{.8in} 
	\noindent {\raggedright{\bf \Large \title}
	\vspace*{.2in}
	
	\hspace*{1in}{\bf \course -- \semester}
	
	\vspace*{.05in}
	
	\hspace*{1in}{\bf  Professor Hohn} 
}
\vspace*{.2in}

% Numbered problems for homework
\begin{enumerate}

\item  Let $g,r,s$ be functions, and let $f$ be a bijective function.  Prove the following: 
	\begin{enumerate}
		\item $f\circ g=f\circ h \implies g=h \quad \mbox{ (left cancellation)}$,

% Insert proof below
		\begin{proof}
			
		\end{proof}
		
		\item  $r\circ f=s\circ f \implies r=s \quad \mbox{ (right cancellation)}$.
		
% Insert proof below
		\begin{proof}
			
		\end{proof}
	\end{enumerate}


\item Suppose that $A$ and $B$ are non-empty sets. Let $f : A \to B$. Given any subset $V \subseteq B$, the \emph{inverse image} of $V$ with respect to $f$ is the set
\[
	f^{-1}(V) = \{ x \in A \mid f(x) \in V\} \subseteq A.
\]
The notation $f^{-1}$ in this context does \emph{not} imply that $f$ is invertible, but it is a convenient and widely used notation for the inverse image of $V$;  this definition makes sense for any function.
	\begin{enumerate}
		\item * Let $V \subseteq B$ and $U = f^{-1}(V)$. Prove that $f(U) = V$ if and only if $V \subseteq f(A)$.
		
% Insert proof below 
		\begin{proof}
			
		\end{proof}
		
	\end{enumerate}


\item (pg. 154 \# 37) If $f:X\to Y$ is a bijective function, prove that its inverse is unique.

% Insert proof below
		\begin{proof}
			
		\end{proof}


\end{enumerate}

\end{document}