Search results
Results from the WOW.Com Content Network
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.
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 .
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 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.
Given a collection of sets, consider the Cartesian product = of all sets in the collection. The canonical projection corresponding to some Y ∈ S {\displaystyle Y\in S} is the function p Y : X → Y {\displaystyle p_{Y}:X\to Y} that maps every element of the product to its Y {\displaystyle Y} component.
The arity of a relation (or predicate) is the dimension of the domain in the corresponding Cartesian product. (A function of arity n thus has arity n +1 considered as a relation.) In computer programming , there is often a syntactical distinction between operators and functions ; syntactical operators usually have arity 1, 2, or 3 (the ternary ...
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 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.