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

Cartesian product is not associative

Formulation 0

Let $X : = \{ 0 \}$, $Y : = \{ 1 \}$, and $Z : = \{ 1 \}$ each be a D11: Set.

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

Remarks

▶	Remark 0 Even though not equal, a weaker result [[[r,4060]]] shows that the sets, however, are isomorphic.

Proofs

Proof 0

Let $X : = \{ 0 \}$, $Y : = \{ 1 \}$, and $Z : = \{ 1 \}$ each be a D11: Set.

By direct computation \begin{equation} \begin{split} X \times (Y \times Z) & = \{ 0 \} \times (\{ 1 \} \times \{ 1 \}) \\ & = \{ 0 \} \times \{ (1, 1) \} \\ & = \{ (0, (1, 1)) \} \\ & \neq \{ ((0, 1), 1) \} \\ & = \{ (0, 1) \} \times \{ 1 \} \\ & = (\{ 0 \} \times \{ 1 \}) \times \{ 1 \} \\ & = (X \times Y) \times Z \\ \end{split} \end{equation} $\square$