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\title{cmpe220 hw \#3}
\author{Your Name}
\date{\today}

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\maketitle

\begin{enumerate}

    \item 
    Consider big-O notation $\mathcal{O}(f)$. 
    Let $\mathbb{F}$ be the set of functions from $\mathbb{Z}^{+}$ to $\mathbb{Z}^{+}$.
    Define relation $\alpha \subseteq \mathbb{F} \times \mathbb{F}$ as
    $f_{1} \alpha f_{2}$ iff $\mathcal{O}(f_{1}) = \mathcal{O}(f_{2})$.
    
    Discuss if $\alpha$ is reflexive, symmetric, antisymmetric, transitive.

    \item 
    \textbf{def}
    A family of subsets $A_{1}, A_{2}, \dotsc, A_{n}$ of a set $A$ is called a \emph{cover} of $A$ 
    iff
    \[
        A_{1} \cup A_{2} \cup \cdots \cup A_{n} = A.
    \]
    
    Show that every cover $C$ of $A$ induces a compatibility relation $\alpha$ on $A$
    such that $a \alpha b$ iff $\{ a, b \} \subseteq A_{i}$, for some $i$, $1 \le i \le n$.
    
\end{enumerate}

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