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What is the multiplication principle in set theory?
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If you have two finite sets $A$ and $B$, where $A$ has $M$ elements and $B$ has $N$ elements, then $A \times B$ has $M \times N$ elements. This rule is called the multiplication principle and is very useful in counting the numbers of elements in sets. The number of elements in a set is denoted by $|A|$, so here we write $|A|=M,|B|=N$, and $|A \times B|=M N$. In the above example, $|A|=3,|B|=2$, thus $|A \times B|=3 \times 2=6$. We can similarly define the Cartesian product of $n$ sets $A_{1}, A_{2}, \cdots, A_{n}$ as
$$A_{1} \times A_{2} \times A_{3} \times \cdots \times A_{n}=\left\{\left(x_{1}, x_{2}, \cdots, x_{n}\right) \mid x_{1} \in A_{1} \text { and } x_{2} \in A_{2} \text { and } \cdots x_{n} \in A_{n}\right\}$$
The multiplication principle states that for finite sets $A_{1}, A_{2}, \cdots, A_{n}$, if
$$\left|A_{1}\right|=M_{1},\left|A_{2}\right|=M_{2}, \cdots,\left|A_{n}\right|=M_{n}$$
then
$$\left|A_{1} \times A_{2} \times A_{3} \times \cdots \times A_{n}\right|=M_{1} \times M_{2} \times M_{3} \times \cdots \times M_{n}$$
An important example of sets obtained using a Cartesian product is $\mathbb{R}^{n}$, where $n$ is a natural number. For $n=2$, we have
\begin{aligned} \mathbb{R}^{2} &=\mathbb{R} \times \mathbb{R} \\ &=\{(x, y) \mid x \in \mathbb{R}, y \in \mathbb{R}\} \end{aligned}
$\mathbb{R}^{3}=\mathbb{R} \times \mathbb{R} \times \mathbb{R}$ and so on.
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