% LESSON FOUR
          
\documentclass{amsart}
\thispagestyle{empty}
\theoremstyle{definition}
\newtheorem*{notation}{Notation}
\begin{document}
             
\begin{center}
	\textbf{Lesson Four: Set Notation, Greek and Hebrew Letters}
\end{center}  

A fringe benefit of being a math major is that you will soon learn 
much of the Greek alphabet and even some Hebrew. Plus,
math is a language unto itself.\\

Let $S=\{\pi,\delta,\Delta\}$ and $T=\{\Omega\}$. 
% To make a { or } appear in the typeset .dvi file, a backslash 
%   must precede the brace here.
% The same is true for $, %, &, and other special LaTeX characters.
% Alternatives exist, such as \lbrace instead of \{.


Then $S \cup T = \{\pi,\delta,\Delta,\Omega\}$, $S \cap T = \emptyset $, 
and $S \times T = \{(\pi,\Omega),(\delta,\Omega),(\Delta,\Omega)\}$.\\
% Do not use the letter x for Cartesian products.

In set notation the interval $(1,3]$ is written $\{x\in\mathbb{R} : 1<x\leq3\}$.\\

Is infinity a number? No! It's more like a state of mind ... 
But $\mathbb{R}=(-\infty, \infty)$.	

\begin{notation}
We write $A \subseteq B$ if and only if for each $x\in A$, we also have $x\in B$.
\end{notation}

\begin{notation}
We write $A \not\subseteq B$ if there exists $x\in A$ such that $x\not\in B$.
\end{notation}

So, $\mathbb{N}\subseteq\mathbb{R}$ but $\mathbb{R}\not\subseteq\mathbb{N}$.\\

Power sets are lots of fun. 

If $X=\{7,\{e\}\}$, 
then $\mathcal{P}(X)=\{\emptyset, \{7\}, \{\{e\}\}, \{7,\{e\}\}\}$.\\

\begin{notation}
The cardinality of $\mathbb{N}$ is referred to as $\aleph_0$, 
which is read ``aleph naught.''
\end{notation}

\end{document}