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The Cartesian square of a set X is the Cartesian product X 2 = X × X. An example is the 2-dimensional plane R 2 = R × R where R is the set of real numbers: [1] R 2 is the set of all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).
In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) —where a ∈ A and b ∈ B. [5] The class of all things (of a given type) that have Cartesian products is called a Cartesian ...
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
The product X×Y is the Cartesian product of X and Y, and Z Y is the set of all functions from Y to Z. The adjointness is expressed by the following fact: the function f : X × Y → Z is naturally identified with the curried function g : X → Z Y defined by g ( x )( y ) = f ( x , y ) for all x in X and y in Y .
The lexicographic combination of two total orders is a linear extension of their product order, and thus the product order is a subrelation of the lexicographic order. [3] The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions. [7]
In the special case of the category of groups, a product always exists: the underlying set of is the Cartesian product of the underlying sets of the , the group operation is componentwise multiplication, and the (homo)morphism : is the projection sending each tuple to its th coordinate.
In the mathematical field of category theory, the product of two categories C and D, denoted C × D and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors. [1]
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology , which can also be given to a product space and which agrees ...