# Cartesian product

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### Etymology

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

### Noun

Cartesian product (plural Cartesian products)

1. (set theory) The set of all possible pairs of elements whose components are members of two sets. Notation: $X\times Y=\{(x,y)\|x\in X\land y\in Y\}$ .
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: $X\times Y=\{(x_{1},...x_{m},y_{1},...y_{n})\|(x_{1},...x_{m})\in X\land (y_{1},...y_{n})\in Y\}$ .
• 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.