Cartesian Product is not Associative

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 $B$.

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 $\tuple {a, \tuple {b, c} }$.

From Cardinality of Cartesian Product, we have that:
 * $\card {A \times \paren {B \times C} } = \card {\paren {A \times B} \times C}$

and so:
 * $A \times \paren {B \times C} \sim \paren {A \times B} \times C$

where $\sim$ denotes set equivalence.

So it matters little whether $A \times B \times C$ is defined as being $A \times \paren {B \times C}$ or $\paren {A \times B} \times C$, 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

 * Bijection between R x (S x T) and (R x S) x T
 * Equality of Cartesian Products