AxBz

AxBz is a notation used in mathematics to denote the Cartesian product of three sets A, B, and Z, with Z denoting the set of integers. The expression is written as A × B × Z and read as "A cross B cross Z." By definition, AxBz = { (a, b, z) | a ∈ A, b ∈ B, z ∈ Z }. It is a subset of the triple product A × B × Z and represents all possible triples formed by selecting one element from each set.

If A and B are sets of numbers, A × B × Z sits inside a three-dimensional coordinate space (for example R^3 when A, B ⊆ R). The triple product is used in problems that involve two qualitative choices and one numerical parameter, such as counting, combinatorics, and database tuples.

Examples: Let A = [0,1], B = {0,1}, and Z be the integers. Then (0.5, 1, 7) ∈ A × B × Z.

Properties: If A1 ⊆ A2, B1 ⊆ B2, and Z1 ⊆ Z2, then A1 × B1 × Z1 ⊆ A2 × B2 × Z2. Cardinality: If A, B, and Z are finite, |A × B × Z| = |A| · |B| · |Z|; otherwise the product is infinite.

History: The Cartesian product concept is named after René Descartes, arising from the development of coordinate geometry and later formalized in set theory.

See also: Cartesian product, n-ary product, tuples.