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Cartesian product of the sets {x,y,z} and {1,2,3}In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. [1]
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]
The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that requires the axiom of choice and is equivalent to it in its most general formulation, [3] and shows why the product topology may be considered the more useful ...
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 ...
7.2.3.1 Incorrectly distributing by swapping ⋂ and ... and binary Cartesian product ... Given any , the following are ...
A set equipped with a total order is a totally ordered set; [5] the terms simply ordered set, [2] linearly ordered set, [3] [5] toset [6] and loset [7] [8] are also used. The term chain is sometimes defined as a synonym of totally ordered set, [5] but generally refers to a totally ordered subset of a given partially ordered set.
An alternative basis for the product topology can be given in terms of trees. The basic open sets can be characterized as: If a finite sequence of natural numbers {w i : i < n} is selected, then the set of all infinite sequences of natural numbers that have value w i at position i for all i < n is a basic open set. Every open set is a countable ...
The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B. Set is thus a topos (and in particular cartesian closed and exact in the sense of Barr). Set is not abelian, additive nor preadditive. Every non-empty set is an injective object in Set.