\section{Common Sets and Cartesian Products}

\definition{}
There are a few sets we use often. They have special names:
\begin{itemize}
	\item $\mathbb{N} = \{0, 1, 2, 3, ...\}$ is the set of \textit{natural numbers}.
	\item $\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$ is the set of \textit{integers}.
	\item $\mathbb{Q}$ is the set of \textit{rational numbers}.
	\item $\mathbb{R}$ is the set of \textit{real numbers}.
\end{itemize}
\note[Note]{$\mathbb{Z}$ is called \say{blackboard zee} or \say{big zee.} Naturally, $\mathbb{N}$, $\mathbb{Q}$, and $\mathbb{R}$ have similar names. \\ This, of course, depends on context. Sometimes \say{zee} is all you need.}


\problem{}
Which of the following sets contain 100? \par
\hint{There may be more than one answer in all the problems below.}

\begin{tcolorbox}[
	colback=white,
	colframe=black,
	width=0.5\textwidth,
	toprule=0.3mm,
	bottomrule=0.3mm,
	leftrule=0.3mm,
	rightrule=0.3mm,
]
	\hfill $\mathbb{N}$ \hfill $\mathbb{Z}$ \hfill $\mathbb{Q}$ \hfill $\mathbb{R}$ \hfill\null
\end{tcolorbox}
\vfill


\problem{}
Which of the following sets contain {\large $\frac{1}{2}$}? \par

\begin{tcolorbox}[
	colback=white,
	colframe=black,
	width=0.5\textwidth,
	toprule=0.3mm,
	bottomrule=0.3mm,
	leftrule=0.3mm,
	rightrule=0.3mm,
]
	\hfill $\mathbb{N}$ \hfill $\mathbb{Z}$ \hfill $\mathbb{Q}$ \hfill $\mathbb{R}$ \hfill\null
\end{tcolorbox}
\vfill



\problem{}
Which of the following sets contain $\pi$? \par

\begin{tcolorbox}[
	colback=white,
	colframe=black,
	width=0.5\textwidth,
	toprule=0.3mm,
	bottomrule=0.3mm,
	leftrule=0.3mm,
	rightrule=0.3mm,
]
	\hfill $\mathbb{N}$ \hfill $\mathbb{Z}$ \hfill $\mathbb{Q}$ \hfill $\mathbb{R}$ \hfill\null
\end{tcolorbox}
\vfill


\problem{}
Which of the following sets contain $\sqrt{-1}$? \par

\begin{tcolorbox}[
	colback=white,
	colframe=black,
	width=0.5\textwidth,
	toprule=0.3mm,
	bottomrule=0.3mm,
	leftrule=0.3mm,
	rightrule=0.3mm,
]
	\hfill $\mathbb{N}$ \hfill $\mathbb{Z}$ \hfill $\mathbb{Q}$ \hfill $\mathbb{R}$ \hfill\null
\end{tcolorbox}

\vfill
\pagebreak


\definition{}
Consider the sets $A$ and $B$. The set $A \times B$ consists of all ordered\footnotemark{} pairs $(a, b)$ where $a \in A$ and $b \in B$. \par
This is called the \textit{cartesian product}, and is usually pronounced \say{$A$ cross $B$}.

\footnotetext{This means that order matters. $(a, b) \neq (b, a)$.}

\vspace{2mm}


For example, $\{1, 2, 3\} \times \{\heartsuit, \star\} = \{(1,\heartsuit),~ (1, \star),~ (2,\heartsuit),~ (2, \star),~ (3,\heartsuit),~ (3, \star)\}$ \par


You can think of this as placing the two sets \say{perpendicular} to one another:

\begin{center}
\begin{tikzpicture}[
	scale=1,
	bullet/.style={circle,inner sep=1.5pt,fill}
]
	\draw[->] (-0.2,0) -- (4,0) node[right]{$A$};
	\draw[->] (0,-0.2) -- (0,3) node[above]{$B$};

	\draw (1,0.1) -- ++ (0,-0.2) node[below]{$1$};
	\draw (2,0.1) -- ++ (0,-0.2) node[below]{$2$};
	\draw (3,0.1) -- ++ (0,-0.2) node[below]{$3$};

	\draw (0.1, 1) -- ++ (-0.2, 0) node[left]{$\heartsuit$};
	\draw (0.1, 2) -- ++ (-0.2, 0) node[left]{$\star$};

	\node[bullet] at (1, 1){};
	\node[bullet] at (2, 1) {};
	\node[bullet] at (3, 1) {};
	\node[bullet] at (1, 2) {};
	\node[bullet] at (2, 2) {};
	\node[bullet] at (3, 2) {};


	\draw[rounded corners] (0.5, 0.5) rectangle (3.5, 2.5) {};
	\node[above] at (2, 2.5) {$A \times B$};

\end{tikzpicture}
\end{center}

\problem{}
Let $A = \{0, 1\} \times \{0, 1\}$ \par
Let $B = \{ a, b\}$ \par
What is $A \times B$?

\vfill

\problem{}
What is $\mathbb{R} \times \mathbb{R}$? \par
\hint{Use the \say{perpendicular} analogy}

\vfill
\pagebreak

\definition{}
$\mathbb{R}^n$ is the set of $n$-tuples of real numbers. \par
In English, this means that an element of $\mathbb{R}^n$ is a list of $n$ real numbers: \par

\vspace{4mm}

Elements of $\mathbb{R}^2$ look like $(a, b)$, where $a, b \in \mathbb{R}$. \hfill \note{\textit{Note:} $\mathbb{R}^2$ is pronounced \say{arrgh-two.}}
Elements of $\mathbb{R}^5$ look like $(a_1, a_2, a_3, a_4, a_5)$, where $a_n \in \mathbb{R}$. \par

$\mathbb{R}^1$ and $\mathbb{R}$ are identical.

\vspace{4mm}

Intuitively, $\mathbb{R}^2$ forms a two-dimensional plane, and $\mathbb{R}^3$ forms a three-dimensional space. \par
$\mathbb{R}^n$ is hard to visualize when $n \geq 4$, but you are welcome to try.

\problem{}
Convince yourself that $\mathbb{R} \times \mathbb{R}$ is $\mathbb{R}^2$. \par
What is $\mathbb{R}^2 \times \mathbb{R}$?
\vfill

\problem{}
What is $\mathbb{N}^2$?
\vfill

\problem{}
What is $\mathbb{Z}^3$?

\vfill
\pagebreak