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\Large Hwk 1
\end{center}

{\Large M366} \hfill {\Large Spring 2017} \\


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\section*{Equivalence Relations}



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\begin{enumerate}
\item Give an example of an equivalence relation.
\item Give an example of a surjective function.
\item Give an example of a partition of a set.
\item Let $X$ be the set of students of our class. Does the rule

\textit{x $\sim$ y if $x$ and $y$ are born in the same year}

define an equivalence relation?
\item Let $X=\{ 1,2,3,4,5,6,7\}$. For each of (a), (b) and (c) do the subsets:
$$(a) \ \ \ Y_1= \{ 1,7\},Y_2= \{ 2,3,4\},Y_3= \{ 6\},
Y_4= \{5\}$$
$$(b) \ \ \ Z_1= \{ 1,7\},Z_2= \{ 2,3,4\},Z_3= \{ 6\}$$
$$(c) \ \ \ W_1= \{ 1,2,5,7\},W_2= \{ 2,3,4,5,6\}$$
define an equivalence relation on $X$? How many elements does the quotient set have? Is $[2] =[3]$? 

\item Let $X=\{Cat, Dog, Mouse, Sheep, Ram, Horse, Kangaroo\}$. \\
Define an equivalence relation on $X$ as follows:  two elements of $X$ are equivalent if they have the same number of letters.

\noindent
\textbf{Comments:}\\
You don't have to prove that this rule is an equivalence relation. You can assume I gave you a good one! Also, if you want to save some time in writing you can just use the first letter to indicate each animal.
\begin{enumerate}
\item How many elements does the quotient set $ X/\sim$ have?
\item Write down the equivalence classes for this equivalence relation.
\item Write down the partition that this equivalence relation determines for the set $X$.
\item Write down explicitly the projection function $p:X \to X/\sim$.

\end{enumerate}
\item Let $X= \mathbb{Z}$ and $Y= \{0,1,2,3,4\}$ and define a surjective function:
$$
rem: X  \to Y
$$ 
by $rem(x)$= the remainder of division of $x$ by $5$.
Describe the equivalence relation induced by $rem$ on the set $X$. Describe the natural bijection between $Y$ and the quotient set $X/\sim$.

\end{enumerate}

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