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\begin{center}\textbf{Problem Set 6 \hspace{.7in} Foundations \hspace{.7in} Due 3/21/16} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}
\item If $f: A \rightarrow B$ and $g: B \rightarrow C$ are injective functions, then $g \circ f$ is injective.

\item If $f: A \rightarrow B$ and $g: B \rightarrow C$ are surjective functions, then $g \circ f$ is surjective.

\item If $f: A \rightarrow B$ and $g: B \rightarrow C$ are functions, and $g \circ f$ is injective, then $g$ is injective.

\item If $f: A \rightarrow B$ and $g: B \rightarrow C$ are functions, and $g \circ f$ is surjective, then $f$ is surjective.

\item If $f: A \rightarrow B$ has an inverse function $g$, then $g$ has $f$ as an inverse function also.

\item Let $f: A \rightarrow B$ be a bijective function.  Then there exists an inverse function $g$ for $f$.

\item Let $f: A \rightarrow B$ be a bijective function.  Then the inverse function of $f$ is unique, i.e. if $g_1$ and $g_2$ are both inverse functions for $f$, then $g_1 = g_2$.

\item Let $f: A \rightarrow B$ be an invertible function.  Then $f$ is bijective.

\end{enumerate}

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