Cartesian Product

# 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.