Bön

B×N denotes the Cartesian product of two sets, B and N. It consists of all ordered pairs (b, n) where b belongs to B and n belongs to N. If B and N are finite with sizes |B| and |N|, then the product has |B|·|N| elements.

In algebra, B and N may carry structures such as groups, rings, or modules. Their direct product B × N is formed by pairing elements and defining the operations componentwise, for example (b1, n1) ⋅ (b2, n2) = (b1 ⋅ b2, n1 ⋅ n2) when both components have compatible operations. The direct product inherits the relevant algebraic structure from its factors, and projections π_B: B×N → B and π_N: B×N → N retrieve the respective components. The universal property states that for any set X with functions f: X→B and g: X→N, there is a unique function X→B×N given by x↦(f(x), g(x)).

In topology, if B and N are topological spaces, B×N is given the product topology. Convergence and continuity are characterized coordinatewise in this setting. If B and N are finite index sets, a B×N matrix or array has B rows and N columns, and is used to represent grids, contingency tables, or data matrices.

Examples: B = {red, blue} and N = {1, 2, 3} yield six elements in B×N. The notation B×N is distinct from B·N (product in a given algebraic category) and from other uses where dimensions are expressed as B-by-N in matrices or grids. See also Cartesian product and direct product.