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\title{Math 214 -- Foundations of Mathematics\\
Homework 8\\
{\large{\bf Due Nov 2, 2011}}}
\date{}
\author{Your name}


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Solve the following problems.  Show all your work.   Four points each. 

\begin{enumerate}

\item (4 points) Let $S$ be a nonempty subset of $\Z$, and let $R$ be a relation defined on $S$ by $(x, y)\in R$ if $3 | (x+2y)$.  
\begin{enumerate}
\item Prove that $R$ is an equivalence relation.  
\item If $S=\{-7, -6, -2, 0, 1, 4, 5, 7\}$, then what are the distinct equivalence classes in this case?\\
\end{enumerate}

\item (4 points) Let $S$ be a non-empty subset of $\N$, and let $\sim$ be a relation defined on $S$ by $x\sim y$ if $x^2+y^2$ is even.  Prove that $\sim$ is an equivalence relation. Determine the distinct equivalence classes.  \\

\item (4 points) Show that the relation $R$ defined on $\R\times \R$ by $((a,b), (c,d))\in \R$ if $|a|+|b|=|c|+|d|$ is an equivalence relation.  describe geometrically the elements of the equivalence classes $[(1,2)], [(3,0)]$. 


\item (4 points) For some nonempty set $S$, suppose $f: S \rightarrow S$ is a function and an equivalence relation.  What is $f$? Justify your answer.  

\item (4 points) Define the mapping $h: \Z_{20}\to \Z_{20}$  by $h([a])=[3a]$ for each $a\in \Z$.  Prove that $h$ is well-defined and  injective. 

\item (4 points) Let $p$ be a positive prime number and $f: \Z_p\to \Z_p$ be defined as $f([x])=[x^2]$.  Show that $f$ is a function.  Give examples to show that it is not necessarily injective or surjective.  
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