87 lines
2.7 KiB
TeX
87 lines
2.7 KiB
TeX
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\section{Notation and Terminology}
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\definition{}
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\begin{itemize}
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\item $\mathbb{R}$ is the set of all real numbers.
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\item $\mathbb{R}^+$ is the set of positive real numbers. Zero is not positive.
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\item $\mathbb{R}^+_0$ is the set of positive real numbers and zero.
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\end{itemize}
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Mathematicians are often inconsistent with their notation. Depending on the author, their mood, and the phase of the moon, $\mathbb{R}^+$ may or may not include zero. I will use the definitions above.
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\definition{}
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Consider two sets $A$ and $B$. The set $A \times B$ consists of all tuples $(a, b)$ where $a \in A$ and $b \in B$. \\
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For example, $\{1, 2, 3\} \times \{\heartsuit, \star\} = \{(1,\heartsuit), (1, \star), (2,\heartsuit), (2, \star), (3,\heartsuit), (3, \star)\}$ \\
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This is called the \textit{cartesian product}.
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\vspace{4mm}
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You can think of this as placing the two sets \say{perpendicular} to one another:
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\begin{center}
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\begin{tikzpicture}[
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scale=1,
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bullet/.style={circle,inner sep=1.5pt,fill}
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]
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\draw[->] (-0.2,0) -- (4,0) node[right]{$A$};
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\draw[->] (0,-0.2) -- (0,3) node[above]{$B$};
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\draw (1,0.1) -- ++ (0,-0.2) node[below]{$1$};
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\draw (2,0.1) -- ++ (0,-0.2) node[below]{$2$};
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\draw (3,0.1) -- ++ (0,-0.2) node[below]{$3$};
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\draw (0.1, 1) -- ++ (-0.2, 0) node[left]{$\heartsuit$};
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\draw (0.1, 2) -- ++ (-0.2, 0) node[left]{$\star$};
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\node[bullet] at (1, 1){};
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\node[bullet] at (2, 1) {};
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\node[bullet] at (3, 1) {};
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\node[bullet] at (1, 2) {};
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\node[bullet] at (2, 2) {};
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\node[bullet] at (3, 2) {};
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\draw[rounded corners] (0.5, 0.5) rectangle (3.5, 2.5) {};
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\node[above] at (2, 2.5) {$A \times B$};
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\end{tikzpicture}
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\end{center}
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\problem{}
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Let $A = \{0, 1\} \times \{0, 1\}$ \\
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Let $B = \{ a, b\}$ \\
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What is $A \times B$?
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\vfill
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\problem{}
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What is $\mathbb{R} \times \mathbb{R}$? \\
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\hint{Use the \say{perpendicular} analogy}
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\vfill
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\pagebreak
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\definition{}
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$\mathbb{R}^n$ is the set of $n$-tuples of real numbers. \\
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In english, this means that an element of $\mathbb{R}^n$ is a list of $n$ real numbers: \\
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\vspace{4mm}
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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.}}
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Elements of $\mathbb{R}^5$ look like $(a_1, a_2, a_3, a_4 a_5)$, where $a_n \in \mathbb{R}$. \\
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$\mathbb{R}^1$ and $\mathbb{R}$ are identical.
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\vspace{4mm}
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Intuitively, $\mathbb{R}^2$ forms a two-dimensional plane, and $\mathbb{R}^3$ forms a three-dimensional space. \\
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$\mathbb{R}^n$ is hard to visualize when $n \geq 4$, but you are welcome to try.
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\problem{}
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Convince yourself that $\mathbb{R} \times \mathbb{R}$ is $\mathbb{R}^2$. \\
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What is $\mathbb{R}^2 \times \mathbb{R}$?
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\vfill
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\pagebreak
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