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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 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 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 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 set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, and are cardinal numbers for every i in I, and < for every i in I, then <. The sum here is the cardinality of the disjoint union of the sets m i, and the product is the cardinality of the Cartesian product.
In mathematics, a finitary relation over a sequence of sets X 1, ..., X n is a subset of the Cartesian product X 1 × ... × X n; that is, it is a set of n-tuples (x 1, ..., x n), each being a sequence of elements x i in the corresponding X i. [1] [2] [3] Typically, the relation describes a possible connection between the elements of an n-tuple.
Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of pairs , i.e. a subset of the Cartesian product A × B of some sets A and B , so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A , B and C .
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 ...