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\title{Math 214 -- Foundations of Mathematics\\
Homework 9\\
{\large{\bf Due Nov 9, 2012}}}
\date{}
\author{your name}

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Solve the following problems.  Show all your work.  Unless otherwise
stated, problems are worth 4 points.


\begin{enumerate}

\item Consider $h: \Z_{16}\to \Z_{24}$ by $h([a])=[3a]$ for each $a\in \Z$. \\
(a) Prove that $h$ is a function. \\
(b) Is $h$ injective? Surjective? Bijective?\\

\item Let $f: \R \times \R \rightarrow \R \times \R$ where, for $(a,b)\in \R$, $f(a,b) = (2a+7, 3b - 3)$.  Prove that $f$ is a bijective function and find $f^{-1}$.\\
    


\item Let $A, B$ and $C$ be nonempty sets and let $f, g$ and $h$ be functions such that $f: A\to B, g: B\to C$ and $h: B\to C$. For each of the following,prove or disprove:\\
(a) if $g\circ f=h\circ f$, then $g=h$.\\
(b) if $f$ is injective and $g\circ f=h\circ f$, then $g=h$. \\



\item For nonempty sets $A, B, C$, let $f: A\to B$ and $g: B\to C$ be functions. \\
(a) Prove that if $g\circ f$ is injective, then $f$ is injective.\\
(b) Disprove that if $g\circ f$ is injective, then $g$ is injective. \\


\item For nonempty sets $A$ and $B$ and functions $f : A \rightarrow
  B$ and $g: B \rightarrow A$ suppose that $g \circ f = i_A$, the
  identity function on $A$.
    \begin{enumerate}
    \item \points{4} Show that $f$ is injective and $g$ is surjective.
    \item \points{2} Show that $f$ is not necessarily surjective.
    \item \points{2} Show that $g$ is not necessarily injective.
    \end{enumerate}
    
    

\item Let $A_1, A_2\subseteq A$. Prove that if $f$ is injective, then $f(A_1)\cap f(A_2)\subseteq f(A_1\cap A_2)$. Give an example that the equality fails. \\



\item (Extra Credit, 4 points) Le $S$ be the set of odd positive integers. A function $F: \N\to S$ is defined by $F(n)=k$ for each $n\in \N$, where $k$ is that odd positive integer for which $3n+1=2^mk$ for some nonnegative integer $m$. Prove or disprove the following: \\
(a) $F$ is injective. \\
(b) $F$ is surjective. 


    
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