U×V

U×V denotes the Cartesian product of two sets U and V. It is the set of all ordered pairs (u, v) where u is an element of U and v is an element of V.

The Cartesian product has several basic properties. It is not generally commutative, meaning U×V is not usually equal to V×U as sets, although there is a natural bijection between them given by (u, v) ↦ (v, u). Two projection maps, π_U: U×V → U and π_V: U×V → V, send a pair (u, v) to its first and second components, respectively. These projections are often used to study the structure of the product.

Cardinality considerations follow simple rules for finite sets: if U and V are finite with sizes m and n, then |U×V| = m·n. In general, the product carries the product of cardinalities, with the appropriate definitions extending to infinite sets.

Example: If U = {1, 2} and V = {a, b}, then U×V = {(1, a), (1, b), (2, a), (2, b)}.

Contexts and variants: In topology, U×V is equipped with the product topology, making it a topological space when U and V are topological spaces. In algebra, the Cartesian product can be given a product structure to form direct products of groups, rings, or vector spaces. The concept also underpins relations, functions, and the graphical representation of mappings as their graphs.