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\title{
    cmpe220 hw \#4
    \\
    Fall 2018
}
\author{Haluk Bingol}
\date{\today}

\begin{document}
\maketitle

\begin{enumerate}

    \item 
    What is wrong with the following argument, 
    which supposedly shows that any relation $R$ on $X$ 
    that is symmetric and transitive is reflexive?
    
    Let $x \in X$. 
    Using symmetry,
    we have $(x, y), (y, x) \in R$,
    by transitivity
    we have $(x, x) \in R$.
    Therefore,
    $R$ is reflexive.

    \item 
    Let $f$ be a function from $X$ to $Y$.
    Let
    \[
        S = \{ f^{-1}(\{ y\}) \ | \ y \in Y \}
    \]
    Show that $S$ is a partition of $X$.
    Describe an equivalence relation that gives rise to this partition.
    
\end{enumerate}

\end{document}