Def/Multiplication of cardinal numbers

 =Commutativity= Suppose that $X$ and $Y$ are sets. Then $\vert X \times Y \vert = \vert Y \times X \vert$, since the switch bijection $sw_{X,Y}$ is a bijection from $X \times Y$ to $Y \times X$. Hence, we find that: $$\vert X \vert \cdot \vert Y \vert = \vert Y \vert \cdot \vert X \vert.$$

In other words, multiplication of cardinal numbers is commutative.

=Associativity= Suppose that $X$, $Y$, and $Z$ are sets. Then $$(\vert X \vert \cdot \vert Y \vert) \cdot \vert Z \vert = \vert X \vert \cdot (\vert Y \vert \cdot \vert Z \vert).$$

This can be seen, using the associative bijection: $$ass_{X,Y,Z} \colon (X \times Y) \times Z \rightarrow X \times (Y \times Z).$$

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