% LESSON TWO
          
\documentclass{amsart}
\thispagestyle{empty}
               
\begin{document}

\begin{center}
\textbf{Lesson Two: Math Blackboard, Logic, Quantifiers and Alignment}
\end{center}  

\textbf{Math Blackboard Symbols}

We use \verb!\mathbb! to display some common sets of numbers.
% \verb is short for 'verbatim' and lets us display .tex
%   commands inline in the .dvi output file.
% The !'s show the start and stop points. You can use a different
%   symbol in place of the ! as long as that symbol doesn't appear
%   in the text you want to display.
          
naturals: $\mathbb{N}$ 
% \mathbb stands for 'math blackboard'.

rationals: $\mathbb{Q}$

reals: $\mathbb{R}$
               
positive reals: $\mathbb{R}^+$\\
             
\textbf{Logic and Quantifiers}

The propositions $P \Rightarrow Q$ and $(\sim P) \vee Q$ are equivalent.

The proposition $R \wedge (\sim R)$ is a contradiction; it is always false.

The negation of $(\forall n\in\mathbb{N})(n \text{ is prime})$ is $(\exists
n\in\mathbb{N})(n \text{ is not prime})$.\\

% Alternative: the command \mbox works in place of \text. 

\textbf{Alignment}
               
We can also do negations step by step. To make things line up nicely, 
we use \LaTeX's built-in ``align'' environment.
% Note that you cannot leave any blank lines in your .tex file between 
%   the start and end of this environment.

\begin{align*}
% The asterisk (*) suppresses the automatic numbering of the following lines.
% You can use \notag to suppress the numbering on any given line.
% Put a \\ at the end of each line EXCEPT the final line.
\sim[(z\text{ is odd })\vee (z\text{ is even})] &\iff
	[\sim(z\text{ is odd })]\wedge [\sim(z\text{ is even})]\\
&\iff (z\text{ is not odd}) \wedge (z\text{ is not even})
\end{align*}
% If you don't like the spacing of the 'not' symbol above, try using 
%   \mathord{\sim} instead of \sim.
% \mathord turns the symbol from a binary relation symbol to an ordinary symbol.


% Time-saving tip: user-defined commands.
% \newcommand{\bbR}{\mathbb{R}} will let you type just \bbR to
%   get \mathbb{R}.
% Though this sort of thing saves time, it may make your .tex file less
%   reader-friendly. For that reason, we will not use them in this manual.

\end{document}