Cartesian product

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Cartesian product

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• 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, of two sets ${\displaystyle X}$ and ${\displaystyle Y}$) The set of all possible ordered pairs of elements, the being first from ${\displaystyle X}$, the second from ${\displaystyle Y}$, written ${\displaystyle X\times Y}$. Formally, the set ${\displaystyle \{(x,y)\;|\;x\in X\;{\text{and}}\;y\in Y\}}$.
2. (databases) All possible combinations of rows between all of the tables listed.
3. (geometry, of an m-dimensional space ${\displaystyle X}$ and an n-dimensional space ${\displaystyle Y}$) An (m+n)-dimensional space, formally composed of all possible ordered pairs of points from ${\displaystyle X}$ and ${\displaystyle Y}$, but thought of as an independent (m+n)-dimensional space (in the sense that if, e.g. ${\displaystyle X}$ and ${\displaystyle Y}$ are vector spaces, the elements of ${\displaystyle X\times Y}$ are thought of as (m+n)-tuples instead of ordered pairs) and written ${\displaystyle X\times Y}$.
   • 1987, M. Göckeler, T. Schücker, Differential Geometry, Gauge Theories, and Gravity, published 1989, page 98:

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

4. (mathematics) Any of several generalizations of the set-theoretic sense, especially one which shares the geometrical intuition outlined above, i.e. one such that the product can be thought of as an object in its own right and not just as a set of pairs.
   • 1997, Michel Marie Deza, Monique Laurent, Geometry of Cuts and Metrics, published 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”, in 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