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If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple.
The product in this category is given by the cartesian product of sets. The coproduct is given by the disjoint union: given sets A i where i ranges over some index set I, we construct the coproduct as the union of A i ×{i} (the cartesian product with i serves to ensure that all the components stay disjoint).
In general, the collections may be indexed over any set I, (called index set whose elements are used as indices for elements in a set) not just R. In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty.
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 ...
The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as Von Neumann–Bernays–Gödel set theory (often called NBG) and Morse–Kelley set theory. The cumulative hierarchy is not compatible with other set theories such as New Foundations.
If G is a group, then the category of all G-sets is Cartesian closed. If Y and Z are two G-sets, then Z Y is the set of all functions from Y to Z with G action defined by (g.F)(y) = g.F(g −1.y) for all g in G, F:Y → Z and y in Y. The subcategory of finite G-sets is also Cartesian closed.
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.
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category.It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms.