Cartesian product

Key properties

• Non‑commutativity: A × B is not the same as B × A unless a specific bijection (swap map) is specified; the order of components matters.
• Cardinality (finite case): If A and B are finite, |A × B| = |A|·|B|.
• Empty factors: If either A or B is empty then A × B is empty.
• Associativity up to isomorphism: There is a natural correspondence between A × (B × C) and (A × B) × C, but the underlying tuples differ in grouping.
• Projection maps: The canonical projections π1: A × B → A and π2: A × B → B send (x,y) to x and y, respectively; these are used to recover components.

The Cartesian product generalizes to n-fold products A1 × A2 × … × An, producing ordered n‑tuples, and to indexed families {Ai}_{i∈I}. For an infinite family, the product consists of all functions f on the index set I with f(i) ∈ Ai for each i. The existence of elements in an infinite product of nonempty sets is related to the axiom of choice: without a specified rule for selecting one element from each Ai, no explicit member may be definable.

Uses and examples

• Coordinate spaces: The real plane R2 is R × R; higher dimensional Euclidean spaces are finite Cartesian products of R.
• Relations and functions: A binary relation between A and B is a subset of A × B; a function from A to B can be viewed as a particular subset of A × B satisfying a uniqueness condition.
• Databases and computing: A Cartesian join (cross join) pairs every row of one table with every row of another, implementing the product construction in practice.
• Algebra and topology: Direct product constructions and the product topology use Cartesian products as their underlying sets.

Several formal points are often emphasized: the conventional set-theoretic definition of ordered pairs (for example the Kuratowski pair) makes the product well defined inside standard frameworks; for indexed empty families the product is commonly identified with the singleton containing the empty function; and many structural features of products are expressed by universal properties in category theory. The Cartesian product is therefore a simple yet pervasive tool that organizes multi‑component data across mathematics and its applications.