Cartesian Product of Sets 📂Set Theory

Cartesian Product of Sets

Definition ¹

For any two objects $a$, $b$, $(a,b)$ is called an Ordered Pair.
For any two sets $A$, $B$, the set of ordered pairs $(a,b)$ of $a \in A$, $b \in B$ is called the Cartesian Product of $A$ and $B$ and is represented as follows. $$ A \times B := \left\{ (a,b): a \in A \land b \in B \right\} $$

Explanation

The reason why the term ‘product’ is used in the Cartesian product comes from considering the number of elements a set has, resulting in $| A \times B | = | A | \times |B|$. For a single set $X$, $X \times X$ is represented as $X^{2}$. If you consider the set of real numbers $\mathbb{R}$, then $\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}$ can be seen as the set containing all points on the Coordinate Plane. This implies that by multiplying the set $\mathbb{R}$, containing all points on the Line, we can obtain the coordinate plane $\mathbb{R}^2$. The term ‘Cartesian’ in Cartesian product is used because René Descartes, who devised this coordinate plane and introduced it to the mathematical world, opening up the world of analytic geometry, is the one attributed with its invention. Although Descartes lived much earlier than Cantor and did not directly contribute to set theory, conceptually, he was ahead, making him fully deserving of this nomenclature’s honor.

Such Cartesian products can naturally be generalized, and as an example, the three-dimensional space $\mathbb{R}^{3}$ as well as the general Euclidean space $\mathbb{R}^{p}, p \in \mathbb{N}$ can be considered. Although it might be hard to imagine, the Cartesian product can be extended beyond natural numbers.

Theorem

Meanwhile, the following distributive laws hold for the Cartesian product.

Distributive Laws

For any sets $A$, $B$, $C$: $$ A \times (B \cap C) = ( A \times B) \cap (A \times C) \\ A \times (B \cup C) = ( A \times B) \cup (A \times C) \\ A \times (B \setminus C) = ( A \times B) \setminus (A \times C) $$