The Cartesian product of sets refers to the collection of all ordered pairs that can be formed from two sets.

More specifically, if we have two sets, which we can denote as Set $A$ and Set $B$, the Cartesian product is represented as $A \times B$. This notation signifies that we take every element from Set $A$ and pair it with every element from Set $B$. For example, if we define Set $A = \{1, 2\}$ and Set $B = \{x, y\}$, then the Cartesian product $A \times B$ would yield the set $\{(1, x), (1, y), (2, x), (2, y)\}$. Each of these pairs is termed an ordered pair, emphasizing that the sequence in which the elements appear is significant; for instance, $(1, x)$ is distinct from $(x, 1)$.

This concept is named after the French philosopher and mathematician René Descartes, who made numerous influential contributions to the field of mathematics. The Cartesian product is a foundational concept in set theory and finds applications across various mathematical domains, including geometry. In geometry, it is instrumental in defining coordinates within a plane. For example, the Cartesian plane is essentially the Cartesian product of the set of real numbers with itself, resulting in all possible pairs of real numbers $(x, y)$.

Grasping the concept of the Cartesian product is valuable as it aids in visualizing and manipulating multi-dimensional data. In computer science, for instance, Cartesian products are utilized in database queries to merge tables. Additionally, you may encounter this concept in everyday situations, often unconsciously, such as when contemplating all possible outfit combinations from a selection of shirts and trousers.