Cartesian product

• cartesian product

From Cartesian + product, after French philosopher, mathematician, and scientist René Descartes (1596–1650), whose formulation of analytic geometry gave rise to the concept.

Cartesian product (plural Cartesian products)

1. (set theory) The set of all possible pairs of elements whose components are members of two sets. Notation: ${\displaystyle X\times Y=\{(x,y)\|x\in X\land y\in Y\}}$ .
2. (databases) All possible combinations of rows between all of the tables listed.
3. (geometry) The set of points in an (m + n)-dimensional Cartesian space corresponding to all possible pairs of points from the two sets from spaces of dimension m and n. Notation: ${\displaystyle X\times Y=\{(x_{1},...x_{m},y_{1},...y_{n})\|(x_{1},...x_{m})\in X\land (y_{1},...y_{n})\in Y\}}$ .
   • 1987, M. Göckeler, T. Schücker, Differential Geometry, Gauge Theories, and Gravity, 1989, page 98,

     On the Cartesian product of two manifolds a differentiable structure can be constructed in the following way.

   • 1997, Michel Marie Deza, Monique Laurent, Geometry of Cuts and Metrics, 2009, page 297,

     The hypercube is the simplest example of a Cartesian product of graphs; indeed, the m-hypercube is nothing but (K₂)^m.

   • 2004, David Bao, Colleen Robles, Ricci and Flag Curvatures in Finsler Geometry, David Dai-Wai Bao, Robert L. Bryant, Shiing-Shen Chern, Zhomgmin Shen (editors), A Sampler of Riemann-Finsler Geometry, page 246,

     A moment's thought convinces us of the following:

     The Cartesian product of two Riemannian Einstein metrics with the same constant Ricci scalar ρ is again Ricci-constant, and has Ric = ρ.

• (set of possible pairs) direct product