Cartesian Product

If \(X\) and \(Y\) are sets, then we define the Cartesian Product \(X \times Y\) to be the collection of ordered pairs, e.g. \((x, y)\), whose first component lies in \(X\) and whose second component lies in \(Y\)

\(X \times Y = \{(x, y) | x \in X, y \in Y\}\)

An example would be \(\mathbb{R} \times \mathbb{R} = \mathbb{R}^{2}\) (i.e. the cartesian plane). \(\mathbb{R}^{2}\) is the cartesian product of “crossing” \(\mathbb{R}\) with itself.