Cartesian Product is not Associative

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

Then:
 * $A \times \left({B \times C}\right) \ne \left({A \times B}\right) \times C$

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

Comment
Despite this result, the cartesian product of three sets is usually just written $A \times B \times C$ and understood to be the set of all ordered triples.

That is, as the set of all elements like $\left({a, \left({b, c}\right)}\right)$.

From Cardinality of Cartesian Product, we have that:
 * $\left|{A \times \left({B \times C}\right)}\right| = \left|{\left({A \times B}\right)\times C}\right|$

and so:
 * $A \times \left({B \times C}\right) \sim \left({A \times B}\right)\times C$

where $\sim$ denotes set equivalence.

So it matters little whether $A \times B \times C$ is defined as being $A \times \left({B \times C}\right)$ or $\left({A \times B}\right)\times C$, and it is rare that one would even need to know.

When absolute rigour is required, the cartesian product of more than two sets can be defined using ordered $n$-tuples or, even more generally, by indexed sets.

Also see

 * Equality of Cartesian Products