Cartesian Product is not Associative/Proof 1

Intuitive Proof
By definition:
 * $A \times B = \left\{{\left({a, b}\right): a \in A, b \in B}\right\}$

that is, the set of all ordered pairs $\left({a, b}\right)$ such that $a \in A$ and $b \in B$.

Now:
 * Elements of $A \times \left({B \times C}\right)$ are in the form $\left({a, \left({b, c}\right)}\right)$
 * Elements of $\left({A \times B}\right)\times C$ are in the form $\left({\left({a, b}\right), c}\right)$.

So for $A \times \left({B \times C}\right) = \left({A \times B}\right)\times C$ we would need to have that $a = \left({a, b}\right)$ and $\left({b, c}\right) = c$.

This can not possibly be so, except perhaps in the most degenerate cases.

So from the strict perspective of the interpretation of the pure definitions, $A \times \left({B \times C}\right) \ne \left({A \times B}\right) \times C$.