ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4632 on D326: Cartesian product
Subresult of R4630: Bijection between parenthesis-sliced cartesian triple products

Cardinality of cartesian triple products is invariant under insertion of parentheses

Formulation 0

Let $X$, $Y$, and $Z$ each be a D11: Set such that

(i)	$X \times Y \times Z$ and $(X \times Y) \times Z$ are each a D326: Cartesian product

Then \begin{equation} |X \times Y \times Z| = |(X \times Y) \times Z| \end{equation}

Proofs

Proof 0

Let $X$, $Y$, and $Z$ each be a D11: Set such that

(i)	$X \times Y \times Z$ and $(X \times Y) \times Z$ are each a D326: Cartesian product