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\date{01/20/2022}
\title{%
        Intro Set Theory\\
            \large MATH--190--01 : Discrete Mathematics for Computing}
\author{Blizzard MacDougall}
\begin{document}
\maketitle
\pagenumbering{arabic}

\section{Sets}
A set is an unordered list. A set is said to contain elements or objects.
Notationally, $a\in S$ is \say{a is contained in S}. $a\notin S$ also exists.
A set is typically an upper case letter, and lower case letters are elements.

The roster method just lists all the elements in a list, with curly braces around them.
If you're going to use an ellipses, you need to give enough info for it to be obvious.
Sets don't \emph{have} to have any common properties, but its helpful if they do.

There's also \emph{set builder notation}. That is, $S={x\in D | P(x)}$, read as
\say{The set $S$, defined as all $x$ contained in $D$ such that $P(x)$}.
Examples:
\begin{equation}
    \begin{split}
    \mathbb{N}=\{0,1,2,3,...\}\\
    \mathbb{Z}=\{...-3,-2,-1,0,1,2,3,...\}\\
    \mathbb{Z}^+=\mathbb{Z}_+=\mathbb{N}+0\\
    \mathbb{Q}=\{\frac{a}{b}|a,b\in \mathbb{Z}\ and\ b\neq0\}\\
    \mathbb{R}=All\ real\ numbers\\
    \mathbb{R}^+=\mathbb{R}_+=\{x\in \mathbb{R}| x> 0\}\\
    \mathbb{C}=all\ complex\ numbers
    \end{split}
\end{equation}
Note that natural numbers are kinda fuzzy. This course says zero is natural, others may not.
Every set is a subset of itself.
Subsets are noted as $A\subseteq B$. i.e. A is a subset of B.
\begin{equation}
    \mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{R}\subseteq\mathbb{Q}\subseteq\mathbb{C}
\end{equation}

Every set has two difinitive subsets, itself, and the empty set (or the null set).$\emptyset$
A subset that is strictly smaller than the superset is called a proper subset.
You can check for this with $\subset$.
Note that there is only one empty set.

Two sets are equal if they contain the same elements. Alternatively, sets are equal
if they contain each other. To show something's unique, show that if there are two of the thing,
then prove they're equal.

\section{Cardinality}
Note that we're sticking with finite sets for now.

If we have a set with exactly $n$ distinct elements, (where $n\in \mathbb{N}$), then
the cardinality of a set is $n$. Cardinality is notationally written as $\#S=|S|=n$.
If $S$ is not finite, its infinite. Cardinality of infinte sets are written with $\aleph_n$.

Notably, $|\emptyset|=0$.
$\wp$

\end{document}
