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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]
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 set X is called the domain of the function [2] and the set Y is called the codomain of the function. [3] Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.
If A and B are sets, then the Cartesian product (or simply product) is defined to be: A × B = {(a,b) | a ∈ A and b ∈ B}. That is, A × B is the set of all ordered pairs whose first coordinate is an element of A and whose second coordinate is an element of B.
The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called the Cartesian product of A and B, and written A × B. A binary relation between sets A and B is a subset of A × B. The (a, b) notation may be used for other purposes, most notably as denoting open intervals on the real number line ...
The Cartesian product of any family of nonempty sets is nonempty. In other words, every family of nonempty sets has a choice function (i.e. a function which maps each of the nonempty sets to one of its elements). König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger ...
6 Cartesian products ⨯ of finitely many sets Toggle Cartesian products ⨯ of finitely many sets subsection 6.1 Binary ⨯ distributes over ⋃ and ⋂ and \ and ∆
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 .