\chapter{Introduction}\label{Chap:Intro}


\section{Examples of Tables}\vspace{-5mm}

\begin{table}[!h]
	\centering
	\caption{The Number of Animals in Zoos of Chiang Mai}
	
	\begin{tabular}{|c|l|r|}
		\hline
		\textbf{No.} & \textbf{Zoos} & \textbf{The Number of Animals}	\\ \hline
		1              & Chiang Mai Zoo                & 1,158                  \\ \hline
		2              & Chiang Mai Night Safari   	& 849	                 \\ \hline
		\multicolumn{2}{|c|}{\textbf{Total}}		& 2,007	                 \\ \hline
	\end{tabular}
	
\end{table}





\section{Examples of figures}\vspace{-5mm}


\begin{figure}[!h]
	\centering
	
	\includegraphics[width=0.5\textwidth]{Graph}
	
	\caption{The Generalized Farey Graph $ \mathcal{G}_{1,2} $}
\end{figure}









\section{Examples of Theorem and Related Environments}

\begin{definition}
	\emph{quadrangle} is a geometric object consisting of four sides and four vertices
\end{definition}

\begin{notation}
	content...
\end{notation}

\begin{example}
	content...
\end{example}

\begin{conclusion}
	content...
\end{conclusion}


\begin{lemma}
	content...
	\begin{proof}
		content...
	\end{proof}
\end{lemma}

\begin{proposition}
	content...
	\begin{proof}[Sketch of Proof]
		content...
	\end{proof}
\end{proposition}

\begin{theorem}
	content...
\end{theorem}

\begin{corollary}
	content...
\end{corollary}

\begin{remark}
	content...
\end{remark}

\begin{conjecture}
	content...
\end{conjecture}






\section{Math Mode in \LaTeX}

\subsection{Inline Math}

It has been known that the equation $ e=mc^2 $ was investigated by 
Albert Einstein.






\subsection{Display Math}

It has been known that the equation 
	\[ e = mc^2 \]
was investigated by Albert Einstein.





It has been known that the equation 
\begin{equation}
	e=mc^2
\end{equation}
was investigated by Albert Einstein.





Suppose that $ ad - bc = 1 $ and $ cz^2 + (d-a)z - b = 0 $. The solutions of the quadratic equation are shown below;	
\begin{align}
	z 	&= \frac{-(d-a) \pm \sqrt{(d-a)^2 - 4c(-b)}}{2c}\\
		&= \frac{(a-d) \pm \sqrt{d^2 + 2ad + a^2 - 4(ad - bc)}}{2c}\\
		&= \frac{(a-d) \pm \sqrt{(d + a)^2 - 4}}{2c}.
\end{align}


\begin{align}
z 	&= \frac{-(d-a) \pm \sqrt{(d-a)^2 - 4c(-b)}}{2c}\\
&= \frac{(a-d) \pm \sqrt{d^2 + 2ad + a^2 - 4(ad - bc)}}{2c}\notag\\
&= \frac{(a-d) \pm \sqrt{(d + a)^2 - 4}}{2c}
\end{align}



\begin{align*}
z 	&= \frac{-(d-a) \pm \sqrt{(d-a)^2 - 4c(-b)}}{2c}\\
&= \frac{(a-d) \pm \sqrt{d^2 + 2ad + a^2 - 4(ad - bc)}}{2c}\\
&= \frac{(a-d) \pm \sqrt{(d + a)^2 - 4}}{2c}\stepcounter{equation}\tag{\theequation}
\end{align*}





\begin{subequations}
	\begin{align}
		z 	&= \frac{-(d-a) \pm \sqrt{(d-a)^2 - 4c(-b)}}{2c}\\
			&= \frac{(a-d) \pm \sqrt{d^2 + 2ad + a^2 - 4(ad - bc)}}{2c}\notag\\
			&= \frac{(a-d) \pm \sqrt{(d + a)^2 - 4}}{2c}
	\end{align}
\end{subequations}




\section{Labeling and Referencing}\label{sec:Lab&Ref}

Section \ref{sec:Lab&Ref} of Chapter \ref{Chap:Intro} describes how we can label and reference our equations.

\subsection{Tables and Figures}


\begin{table}[!h]
	\centering
	\caption{The Number of Animals in Zoos of Chiang Mai}\label{Tab:NoOfAnimal}
	
	\begin{tabular}{|c|l|r|}
		\hline
		\textbf{No.} & \textbf{Zoos} & \textbf{The Number of Animals}	\\ \hline
		1              & Chiang Mai Zoo                & 1,158                  \\ \hline
		2              & Chiang Mai Night Safari   	& 849	                 \\ \hline
		\multicolumn{2}{|c|}{\textbf{Total}}		& 2,007	                 \\ \hline
	\end{tabular}
	
\end{table}


Table \ref{Tab:NoOfAnimal} shows the numbers of animals in Chiang Mai Zoo and Chiang Mai Night Safari.

\begin{figure}[!h]
	\centering
	
	\includegraphics[width=0.5\textwidth]{Graph}
	
	\caption{The Graph $ \mathcal{G}_{1,2} $}\label{fig:G12}
\end{figure}

Figure \ref{fig:G12} demonstrates a part of the graph $ \mathcal{G}_{1,2} $ 




\subsection{Equations}

\begin{equation}\label{eq:emc2}
	e=mc^2
\end{equation}
It has been known that the equation \eqref{eq:emc2} was investigated
by Albert Einstein.



\begin{align}
	z 	&= \frac{-(d-a) \pm \sqrt{(d-a)^2 - 4c(-b)}}{2c} \label{eq:Q1st}\\
		&= \frac{(a-d) \pm \sqrt{d^2 + 2ad + a^2 - 4(ad - bc)}}{2c} \label{eq:QMid}\\
		&= \frac{(a-d) \pm \sqrt{(d + a)^2 - 4}}{2c}. \label{eq:QLast}
\end{align}
\eqref{eq:Q1st} is the quadratic formula. Then we add $ +4ad-4ad $ in the square root and obtain  \eqref{eq:QMid} by regrouping the variables. Finally, replacing $ ad-bc $ by $ 1 $ provides \eqref{eq:QLast}.






\begin{align*}
	z 	&= \frac{-(d-a) \pm \sqrt{(d-a)^2 - 4c(-b)}}{2c} \\
		&= \frac{(a-d) \pm \sqrt{d^2 + 2ad + a^2 - 4(ad - bc)}}{2c} \\
		&= \frac{(a-d) \pm \sqrt{(d + a)^2 - 4}}{2c}. \stepcounter{equation}\tag{\theequation}\label{eq:QLast2}
\end{align*}
\eqref{eq:QLast2} is the simplest form of $ z $.





\subsection{Theorem and Others}

\begin{theorem}\label{thm:FareyChromatic}
	The Farey graph $ \mathcal{F} $ is $ 3 $-chromatic.
\end{theorem}
Theorem \ref{thm:FareyChromatic} is the first result of chromatic numbers for the graphs $ \mathcal{F}^K_{u,n} $.



\begin{theorem}\thlabel{thm:FChromatic}
	The Farey graph $ \mathcal{F} $ is $ 3 $-chromatic.
\end{theorem}
\thref{thm:FChromatic} is the first result of chromatic numbers for the graphs $ \mathcal{F}^K_{u,n} $.


Theorem \ref{thm:FChromatic} is the first result of chromatic numbers for the graphs $ \mathcal{F}^K_{u,n} $.

