Cartesian Product of the Reals — R×R and R×R×R

Taking $A = B = \mathbb{R}$, the set of real numbers, in the definition of the previous section gives one of the most important sets in the whole of mathematics.

$\mathbb{R} \times \mathbb{R}$.

Every ordered pair of real numbers $(x, y)$ is an element of $\mathbb{R} \times \mathbb{R}$, and conversely every element of $\mathbb{R} \times \mathbb{R}$ is such a pair. This is precisely the set that is identified with the Cartesian (rectangular coordinate) plane: fixing a horizontal $x$-axis and a vertical $y$-axis through a common origin, the ordered pair $(x, y)$ is represented by the point obtained by moving $x$ units along the $x$-axis and $y$ units parallel to the $y$-axis. This correspondence — one ordered pair of reals for every point of the plane, and one point for every ordered pair — is why $\mathbb{R} \times \mathbb{R}$ is also written $\mathbb{R}^2$.

$\mathbb{R} \times \mathbb{R} \times \mathbb{R}$. Extending the same idea to ordered triples,

also written $\mathbb{R}^3$, is identified with three-dimensional space in exactly the same way: three mutually perpendicular axes ($x$, $y$, $z$) through a common origin let every ordered triple of reals be represented by a unique point in space, and conversely. So $(2, -1, 3) \in \mathbb{R} \times \mathbb{R} \times \mathbb{R}$ names the point reached by moving $2$ units along the $x$-axis, $-1$ unit along the $y$-axis and $3$ units along the $z$-axis — while a pair such as $(0,0)$, having only two components, belongs to $\mathbb{R}\times\mathbb{R}$ and not to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ at all. …