Are Cartesian Product and Multiplication (kind of) equivalent? Example(not trying to prove anything):
$|\{X, Y, Z\}| \times |\{A, B\}| = |\{XA, XB, YA, YB, ZA, ZB\}| = 3 \cdot 2 = 6.$
 A: One thing is sure: 
$$
\text{card } (A\times B) = \text{card } A \times \text{card } B
$$
This can be proven via induction:
$$
\text{card } (A\times {b}) = \text{card } A
$$
andif $b\notin B$:
\begin{align}
\text{card } (A\times (B\cup {b})) &= 
\text{card } ((A\times B) \cup (A\times{b})) \\&=
\text{card } (A\times B) + \text{card } (A\times{b}) \\&=
\text{card } A \times \text{card } B + \text{card } A \\&=
A \times (\text{card } B + 1) \\&=
A \times \text{card } (B \cup \{b\}) 
\end{align}

The same way, you have
$$
\text{card } B^A = \text{card} \{ f : A\to B \} = 
(\text{card }B)^{\text{card }A}
$$
A: Certainly there is a similarity, and this explains the choice of notation for the Cartesian product. Another similarity: you can think of the "multiplication table" of X, Y, Z "times" A, B as containing the elements of the Cartesian product between those two sets. From this your observation that the size of the Cartesian product is the product of the sizes of the factors follows directly.  
