\renewcommand{\coddate}{{\small \today}}
\renewcommand{\sheweek}{Elementary set theory}

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\section*{Elementary Set Theory}
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\begin{enumerate}
\item \textbf{Notation}

\begin{tabular}{rll} \hline
          & Set terminology       & Probability terminology \\ \hline \hline
 $\Omega$ & Collection of objects & Sample space \\
 & &  \\
 $\omega$ & Member of $\Omega$    & Outcome \\
 & &  \\
 $A$      & Subset of $\Omega$    & Event - an outcome in $A$ occurs\\
 & &  \\
 $A^c$    & Complement            & Event - an outcome not in $A$ occurs\\
 & &  \\
 $A \cap B$ & Intersection        & Event - an outcome in $A$ and $B$ occurs\\
 & &  \\
 $A \cup B$ & Union               & Event - an outcome in $A$ and/or $B$ occurs \\
 & &  \\
 $A \backslash B $  & Difference  & Event - an outcome in $A$ but not in $B$ occurs\\
 & &  \\
 $A \subseteq B$ & Inclusion      & if outcome in $A$ also in $B$\\
 & &  \\
 $\emptyset$ & Empty set          & Impossible event\\
 & &  \\
 $\Omega$    & Whole space        & Certain event\\
 & &  \\
 \hline
\end{tabular}
\normalsize

\vspace{0.2cm}
\item \textbf{Properties of Intersections and Unions} \newline
\begin{tabular}{lll}\hline
Commutative &  $A \cap B = B \cap A$ & $A \cup B = B \cup A$ \\
Associative &  $A \cap (B \cap C) = (A \cap B) \cap C$ & $A \cup (B \cup C) = (A \cup B) \cup C$ \\
Distributive & $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ & $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ \\
With whole space & $A \cap \Omega = A $ & $A \cup \Omega = \Omega $ \\
With empty set & $A \cap \emptyset = \emptyset$ & $A \cup \emptyset = A$ \\ \hline
\end{tabular}

\item \textbf{Complement of $A$}: is $A^c = \Omega
\backslash A$, that is, $\omega \in A^c \Longleftrightarrow \omega
\notin A$.
\begin{enumerate}
\item $(A^c)^c = A$.
\item $A \cap A^c = \emptyset$.
\item $A \cup A^c = \Omega$.
\item $(A \cup B)^c = A^c \cap B^c$. \label{deminit:pt}
\end{enumerate}

\item \textbf{Collection of subsets of $\Omega$}:
say $\{A_1, \ldots, A_n\}$, where $n$ is not necessarily finite.
\begin{enumerate}
\item Mutually exclusive: if $A_i
\cap A_j = \emptyset$ for any $i \neq j$, so $A_1,\ldots,A_n$ are
disjoint sets.
\item Exhaustive: if $\bigcup_{i=1}^n A_i =
\Omega$.
\item De Morgan's theorem (a generalization of \ref{deminit:pt}
above): $(\bigcup_{i=1}^n A_i)^c = \bigcap_{i=1}^n A_i^c$.
\end{enumerate}
\end{enumerate}
\newpage
