Proposition 2

Let A and B be two sets. The relation $A \times B = \emptyset$ is equivalent to "$A = \emptyset$ or $B = \emptyset$".

For the relation $z \in A \times B$ implies $\mathrm{pr}_1 z \in A$ and $\mathrm{pr}_2 z \in B$; hence $A \neq \emptyset$ and $B \neq \emptyset$. Conversely, the relation "$x \in A$ and $y \in B$" implies $(x, y) \in A \times B$ and hence $A \times B \neq \emptyset$. In other words, the relation $A \times B \neq \emptyset$ is equivalent to "$A \neq \emptyset$ and $B \neq \emptyset$"; hence the result.

If A, B, C are sets, we write

an element $((x, y), z)$ of $A \times B \times C$ is written $(x, y, z)$ and is called a triple. Again, if A, B, C, D are sets, we write

and so on.