### Cartesian product definition

The Cartesian product $X \times Y$ between two sets $X$ and $Y$ is the set of all possible ordered pairs with first element from $X$ and second element from $Y$: $$X \times Y = \{ (x,y): x \in X \text{ and } y \in Y \}.$$

One example is the standard Cartesian coordinates of the plane, where $X$ is the set of points on the $x$-axis, $Y$ is the set of points on the $y$-axis, and $X \times Y$ is the $xy$-plane.

If $X=Y$, we can denote the Cartesian product of $X$ with itself as $X \times X = X^2$. For examples, since we can represent both the $x$-axis and the $y$-axis as the set of real numbers $\R$, we can write the $xy$-plane as $\R \times \R = \R^2$.