# Cartesian Product

##### Updated: October 21, 2020

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.