Cartesian Product is not Associative/Proof 1

Theorem

Let $A, B, C$ be non-empty sets.

Then:

$A \times \paren {B \times C} \ne \paren {A \times B} \times C$

where $A \times B$ is the cartesian product of $A$ and $B$.

Intuitive Proof

By definition:

$A \times B = \set {\tuple {a, b}: a \in A, b \in B}$

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

Now:

Elements of $A \times \paren {B \times C}$ are in the form $\tuple {a, \tuple {b, c} }$

Elements of $\paren {A \times B} \times C$ are in the form $\tuple {\tuple {a, b}, c}$.

So for $A \times \paren {B \times C} = \paren {A \times B} \times C$ we would need to have that $a = \tuple {a, b}$ and $\tuple {b, c} = c$.

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:

$A \times \paren {B \times C} \ne \paren {A \times B} \times C$

$\blacksquare$