Cartesian product

Lore & Background

The Cartesian product is named after René Descartes, whose work in analytic geometry introduced the idea of representing points in a plane by ordered pairs of real numbers. This allowed geometrical shapes to be described numerically. The set of all such pairs, ℝ × ℝ, corresponds to all points in the plane. The concept was later formalized in set theory, where the Cartesian product of two sets A and B is defined as the set of all ordered pairs (a, b) with a in A and b in B. A table can be created by taking the Cartesian product of a set of rows and a set of columns, with cells containing ordered pairs. The n-fold Cartesian product can be represented by an n-dimensional array.

Reader's Guide

The Cartesian product is a cornerstone of set theory and mathematics, providing a rigorous way to combine sets into ordered pairs and tuples. Its definition relies on the set-theoretic definition of ordered pairs, most commonly Kuratowski's definition, which represents (x, y) as {{x}, {x, y}}. This allows the Cartesian product to be constructed within ZFC set theory using the axioms of pairing, union, power set, and specification. The product is not commutative: A × B equals B × A only if A equals B or one set is empty. It is also not associative: (A × B) × C differs from A × (B × C) unless a set is empty. The Cartesian product interacts with set operations in specific ways; for example, (A ∩ B) × (C ∩ D) equals (A × C) ∩ (B × D), but the analogous property for union does not hold. The concept is essential for defining relations and functions, which are typically subsets of Cartesian products. Its historical origin in Descartes' analytic geometry remains the main illustrative example, where the Cartesian plane is modeled as ℝ × ℝ.

Did You Know?

• The Cartesian product of a 13-element set of ranks and a 4-element set of suits yields the 52 ordered pairs representing a standard deck of playing cards.
• Under Kuratowski's definition, an ordered pair (x, y) is defined as the set {{x}, {x, y}}.
• The Cartesian product A × B is not commutative unless A equals B or one of the sets is empty.