The Cartesian product of two sets and (also called the product set, set direct product, or cross product) is the set of all ordered pairs where and . It is denoted , and is called the Cartesian product since it originated in Descartes' formulation of analytic geometry. In the Cartesian view, points in the plane are specified by their vertical and horizontal coordinates, with points on a line being specified by just one coordinate. The main examples of direct products are Euclidean three-space (, where are the real numbers), and the plane ().

The Cartesian product can be computed in the Wolfram Language using Tuples[{list1, list2}], where list1 and list2 represent the two sets.

The term Cartesian product is also used for the graph Cartesian product (Vizing 1963, Clark and Suen 2000).

See also

Direct Product, Disjoint Union, External Direct Product, External Direct Sum, Graph Cartesian Product, Graph Product, Group Direct Product, Product Set, Product Space

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References

Clark, W. E. and Suen, S. "An Inequality Related to Vizing's Conjecture." Elec. J. Combin. 7, N4, 2000. https://doi.org/10.37236/1542.Comtet, L. "Product Sets." §1.2 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 3-4, 1974.Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 49-50, 1984.Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, p. 3, 1988.Vizing, V. G. "The Cartesian Product of Graphs." Vyčisl. Sistemy 9, 30-43, 1963.

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Cartesian Product

Cite this as:

Weisstein, Eric W. "Cartesian Product." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CartesianProduct.html

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